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1 | 6090-6093 | 7 4
A 60 mF capacitor is connected to a 110 V, 60 Hz ac supply Determine
the rms value of the current in the circuit 7 |
1 | 6091-6094 | 4
A 60 mF capacitor is connected to a 110 V, 60 Hz ac supply Determine
the rms value of the current in the circuit 7 5
In Exercises 7 |
1 | 6092-6095 | Determine
the rms value of the current in the circuit 7 5
In Exercises 7 3 and 7 |
1 | 6093-6096 | 7 5
In Exercises 7 3 and 7 4, what is the net power absorbed by each
circuit over a complete cycle |
1 | 6094-6097 | 5
In Exercises 7 3 and 7 4, what is the net power absorbed by each
circuit over a complete cycle Explain your answer |
1 | 6095-6098 | 3 and 7 4, what is the net power absorbed by each
circuit over a complete cycle Explain your answer 7 |
1 | 6096-6099 | 4, what is the net power absorbed by each
circuit over a complete cycle Explain your answer 7 6
A charged 30 mF capacitor is connected to a 27 mH inductor |
1 | 6097-6100 | Explain your answer 7 6
A charged 30 mF capacitor is connected to a 27 mH inductor What is
the angular frequency of free oscillations of the circuit |
1 | 6098-6101 | 7 6
A charged 30 mF capacitor is connected to a 27 mH inductor What is
the angular frequency of free oscillations of the circuit 7 |
1 | 6099-6102 | 6
A charged 30 mF capacitor is connected to a 27 mH inductor What is
the angular frequency of free oscillations of the circuit 7 7
A series LCR circuit with R = 20 W, L = 1 |
1 | 6100-6103 | What is
the angular frequency of free oscillations of the circuit 7 7
A series LCR circuit with R = 20 W, L = 1 5 H and C = 35 mF is connected
to a variable-frequency 200 V ac supply |
1 | 6101-6104 | 7 7
A series LCR circuit with R = 20 W, L = 1 5 H and C = 35 mF is connected
to a variable-frequency 200 V ac supply When the frequency of the
supply equals the natural frequency of the circuit, what is the average
power transferred to the circuit in one complete cycle |
1 | 6102-6105 | 7
A series LCR circuit with R = 20 W, L = 1 5 H and C = 35 mF is connected
to a variable-frequency 200 V ac supply When the frequency of the
supply equals the natural frequency of the circuit, what is the average
power transferred to the circuit in one complete cycle 7 |
1 | 6103-6106 | 5 H and C = 35 mF is connected
to a variable-frequency 200 V ac supply When the frequency of the
supply equals the natural frequency of the circuit, what is the average
power transferred to the circuit in one complete cycle 7 8
Figure 7 |
1 | 6104-6107 | When the frequency of the
supply equals the natural frequency of the circuit, what is the average
power transferred to the circuit in one complete cycle 7 8
Figure 7 17 shows a series LCR circuit connected to a variable
frequency 230 V source |
1 | 6105-6108 | 7 8
Figure 7 17 shows a series LCR circuit connected to a variable
frequency 230 V source L = 5 |
1 | 6106-6109 | 8
Figure 7 17 shows a series LCR circuit connected to a variable
frequency 230 V source L = 5 0 H, C = 80mF, R = 40 W |
1 | 6107-6110 | 17 shows a series LCR circuit connected to a variable
frequency 230 V source L = 5 0 H, C = 80mF, R = 40 W (a) Determine the source frequency which drives the circuit in
resonance |
1 | 6108-6111 | L = 5 0 H, C = 80mF, R = 40 W (a) Determine the source frequency which drives the circuit in
resonance (b) Obtain the impedance of the circuit and the amplitude of current
at the resonating frequency |
1 | 6109-6112 | 0 H, C = 80mF, R = 40 W (a) Determine the source frequency which drives the circuit in
resonance (b) Obtain the impedance of the circuit and the amplitude of current
at the resonating frequency (c) Determine the rms potential drops across the three elements of
the circuit |
1 | 6110-6113 | (a) Determine the source frequency which drives the circuit in
resonance (b) Obtain the impedance of the circuit and the amplitude of current
at the resonating frequency (c) Determine the rms potential drops across the three elements of
the circuit Show that the potential drop across the LC
combination is zero at the resonating frequency |
1 | 6111-6114 | (b) Obtain the impedance of the circuit and the amplitude of current
at the resonating frequency (c) Determine the rms potential drops across the three elements of
the circuit Show that the potential drop across the LC
combination is zero at the resonating frequency FIGURE 7 |
1 | 6112-6115 | (c) Determine the rms potential drops across the three elements of
the circuit Show that the potential drop across the LC
combination is zero at the resonating frequency FIGURE 7 17
Rationalised 2023-24
Chapter Eight
ELECTROMAGNETIC
WAVES
8 |
1 | 6113-6116 | Show that the potential drop across the LC
combination is zero at the resonating frequency FIGURE 7 17
Rationalised 2023-24
Chapter Eight
ELECTROMAGNETIC
WAVES
8 1 INTRODUCTION
In Chapter 4, we learnt that an electric current produces magnetic field
and that two current-carrying wires exert a magnetic force on each other |
1 | 6114-6117 | FIGURE 7 17
Rationalised 2023-24
Chapter Eight
ELECTROMAGNETIC
WAVES
8 1 INTRODUCTION
In Chapter 4, we learnt that an electric current produces magnetic field
and that two current-carrying wires exert a magnetic force on each other Further, in Chapter 6, we have seen that a magnetic field changing with
time gives rise to an electric field |
1 | 6115-6118 | 17
Rationalised 2023-24
Chapter Eight
ELECTROMAGNETIC
WAVES
8 1 INTRODUCTION
In Chapter 4, we learnt that an electric current produces magnetic field
and that two current-carrying wires exert a magnetic force on each other Further, in Chapter 6, we have seen that a magnetic field changing with
time gives rise to an electric field Is the converse also true |
1 | 6116-6119 | 1 INTRODUCTION
In Chapter 4, we learnt that an electric current produces magnetic field
and that two current-carrying wires exert a magnetic force on each other Further, in Chapter 6, we have seen that a magnetic field changing with
time gives rise to an electric field Is the converse also true Does an
electric field changing with time give rise to a magnetic field |
1 | 6117-6120 | Further, in Chapter 6, we have seen that a magnetic field changing with
time gives rise to an electric field Is the converse also true Does an
electric field changing with time give rise to a magnetic field James Clerk
Maxwell (1831-1879), argued that this was indeed the case – not only
an electric current but also a time-varying electric field generates magnetic
field |
1 | 6118-6121 | Is the converse also true Does an
electric field changing with time give rise to a magnetic field James Clerk
Maxwell (1831-1879), argued that this was indeed the case – not only
an electric current but also a time-varying electric field generates magnetic
field While applying the Ampere’s circuital law to find magnetic field at a
point outside a capacitor connected to a time-varying current, Maxwell
noticed an inconsistency in the Ampere’s circuital law |
1 | 6119-6122 | Does an
electric field changing with time give rise to a magnetic field James Clerk
Maxwell (1831-1879), argued that this was indeed the case – not only
an electric current but also a time-varying electric field generates magnetic
field While applying the Ampere’s circuital law to find magnetic field at a
point outside a capacitor connected to a time-varying current, Maxwell
noticed an inconsistency in the Ampere’s circuital law He suggested the
existence of an additional current, called by him, the displacement
current to remove this inconsistency |
1 | 6120-6123 | James Clerk
Maxwell (1831-1879), argued that this was indeed the case – not only
an electric current but also a time-varying electric field generates magnetic
field While applying the Ampere’s circuital law to find magnetic field at a
point outside a capacitor connected to a time-varying current, Maxwell
noticed an inconsistency in the Ampere’s circuital law He suggested the
existence of an additional current, called by him, the displacement
current to remove this inconsistency Maxwell formulated a set of equations involving electric and magnetic
fields, and their sources, the charge and current densities |
1 | 6121-6124 | While applying the Ampere’s circuital law to find magnetic field at a
point outside a capacitor connected to a time-varying current, Maxwell
noticed an inconsistency in the Ampere’s circuital law He suggested the
existence of an additional current, called by him, the displacement
current to remove this inconsistency Maxwell formulated a set of equations involving electric and magnetic
fields, and their sources, the charge and current densities These
equations are known as Maxwell’s equations |
1 | 6122-6125 | He suggested the
existence of an additional current, called by him, the displacement
current to remove this inconsistency Maxwell formulated a set of equations involving electric and magnetic
fields, and their sources, the charge and current densities These
equations are known as Maxwell’s equations Together with the Lorentz
force formula (Chapter 4), they mathematically express all the basic laws
of electromagnetism |
1 | 6123-6126 | Maxwell formulated a set of equations involving electric and magnetic
fields, and their sources, the charge and current densities These
equations are known as Maxwell’s equations Together with the Lorentz
force formula (Chapter 4), they mathematically express all the basic laws
of electromagnetism The most important prediction to emerge from Maxwell’s equations
is the existence of electromagnetic waves, which are (coupled) time-
varying electric and magnetic fields that propagate in space |
1 | 6124-6127 | These
equations are known as Maxwell’s equations Together with the Lorentz
force formula (Chapter 4), they mathematically express all the basic laws
of electromagnetism The most important prediction to emerge from Maxwell’s equations
is the existence of electromagnetic waves, which are (coupled) time-
varying electric and magnetic fields that propagate in space The speed
of the waves, according to these equations, turned out to be very close to
Rationalised 2023-24
Physics
202
the speed of light( 3 ×108 m/s), obtained from optical
measurements |
1 | 6125-6128 | Together with the Lorentz
force formula (Chapter 4), they mathematically express all the basic laws
of electromagnetism The most important prediction to emerge from Maxwell’s equations
is the existence of electromagnetic waves, which are (coupled) time-
varying electric and magnetic fields that propagate in space The speed
of the waves, according to these equations, turned out to be very close to
Rationalised 2023-24
Physics
202
the speed of light( 3 ×108 m/s), obtained from optical
measurements This led to the remarkable conclusion
that light is an electromagnetic wave |
1 | 6126-6129 | The most important prediction to emerge from Maxwell’s equations
is the existence of electromagnetic waves, which are (coupled) time-
varying electric and magnetic fields that propagate in space The speed
of the waves, according to these equations, turned out to be very close to
Rationalised 2023-24
Physics
202
the speed of light( 3 ×108 m/s), obtained from optical
measurements This led to the remarkable conclusion
that light is an electromagnetic wave Maxwell’s work
thus unified the domain of electricity, magnetism and
light |
1 | 6127-6130 | The speed
of the waves, according to these equations, turned out to be very close to
Rationalised 2023-24
Physics
202
the speed of light( 3 ×108 m/s), obtained from optical
measurements This led to the remarkable conclusion
that light is an electromagnetic wave Maxwell’s work
thus unified the domain of electricity, magnetism and
light Hertz, in 1885, experimentally demonstrated the
existence of electromagnetic waves |
1 | 6128-6131 | This led to the remarkable conclusion
that light is an electromagnetic wave Maxwell’s work
thus unified the domain of electricity, magnetism and
light Hertz, in 1885, experimentally demonstrated the
existence of electromagnetic waves Its technological use
by Marconi and others led in due course to the
revolution in communication that we are witnessing
today |
1 | 6129-6132 | Maxwell’s work
thus unified the domain of electricity, magnetism and
light Hertz, in 1885, experimentally demonstrated the
existence of electromagnetic waves Its technological use
by Marconi and others led in due course to the
revolution in communication that we are witnessing
today In this chapter, we first discuss the need for
displacement current and its consequences |
1 | 6130-6133 | Hertz, in 1885, experimentally demonstrated the
existence of electromagnetic waves Its technological use
by Marconi and others led in due course to the
revolution in communication that we are witnessing
today In this chapter, we first discuss the need for
displacement current and its consequences Then we
present a descriptive account of electromagnetic waves |
1 | 6131-6134 | Its technological use
by Marconi and others led in due course to the
revolution in communication that we are witnessing
today In this chapter, we first discuss the need for
displacement current and its consequences Then we
present a descriptive account of electromagnetic waves The broad spectrum of electromagnetic waves,
stretching from g rays (wavelength ~10–12 m) to long
radio waves (wavelength ~106 m) is described |
1 | 6132-6135 | In this chapter, we first discuss the need for
displacement current and its consequences Then we
present a descriptive account of electromagnetic waves The broad spectrum of electromagnetic waves,
stretching from g rays (wavelength ~10–12 m) to long
radio waves (wavelength ~106 m) is described 8 |
1 | 6133-6136 | Then we
present a descriptive account of electromagnetic waves The broad spectrum of electromagnetic waves,
stretching from g rays (wavelength ~10–12 m) to long
radio waves (wavelength ~106 m) is described 8 2 DISPLACEMENT CURRENT
We have seen in Chapter 4 that an electrical current
produces a magnetic field around it |
1 | 6134-6137 | The broad spectrum of electromagnetic waves,
stretching from g rays (wavelength ~10–12 m) to long
radio waves (wavelength ~106 m) is described 8 2 DISPLACEMENT CURRENT
We have seen in Chapter 4 that an electrical current
produces a magnetic field around it Maxwell showed
that for logical consistency, a changing electric field must
also produce a magnetic field |
1 | 6135-6138 | 8 2 DISPLACEMENT CURRENT
We have seen in Chapter 4 that an electrical current
produces a magnetic field around it Maxwell showed
that for logical consistency, a changing electric field must
also produce a magnetic field This effect is of great
importance because it explains the existence of radio
waves, gamma rays and visible light, as well as all other
forms of electromagnetic waves |
1 | 6136-6139 | 2 DISPLACEMENT CURRENT
We have seen in Chapter 4 that an electrical current
produces a magnetic field around it Maxwell showed
that for logical consistency, a changing electric field must
also produce a magnetic field This effect is of great
importance because it explains the existence of radio
waves, gamma rays and visible light, as well as all other
forms of electromagnetic waves To see how a changing electric field gives rise to
a magnetic field, let us consider the process of
charging of a capacitor and apply Ampere’s circuital
law given by (Chapter 4)
“B |
1 | 6137-6140 | Maxwell showed
that for logical consistency, a changing electric field must
also produce a magnetic field This effect is of great
importance because it explains the existence of radio
waves, gamma rays and visible light, as well as all other
forms of electromagnetic waves To see how a changing electric field gives rise to
a magnetic field, let us consider the process of
charging of a capacitor and apply Ampere’s circuital
law given by (Chapter 4)
“B dl = m0 i (t)
(8 |
1 | 6138-6141 | This effect is of great
importance because it explains the existence of radio
waves, gamma rays and visible light, as well as all other
forms of electromagnetic waves To see how a changing electric field gives rise to
a magnetic field, let us consider the process of
charging of a capacitor and apply Ampere’s circuital
law given by (Chapter 4)
“B dl = m0 i (t)
(8 1)
to find magnetic field at a point outside the capacitor |
1 | 6139-6142 | To see how a changing electric field gives rise to
a magnetic field, let us consider the process of
charging of a capacitor and apply Ampere’s circuital
law given by (Chapter 4)
“B dl = m0 i (t)
(8 1)
to find magnetic field at a point outside the capacitor Figure 8 |
1 | 6140-6143 | dl = m0 i (t)
(8 1)
to find magnetic field at a point outside the capacitor Figure 8 1(a) shows a parallel plate capacitor C which
is a part of circuit through which a time-dependent
current i (t) flows |
1 | 6141-6144 | 1)
to find magnetic field at a point outside the capacitor Figure 8 1(a) shows a parallel plate capacitor C which
is a part of circuit through which a time-dependent
current i (t) flows Let us find the magnetic field at a
point such as P, in a region outside the parallel plate
capacitor |
1 | 6142-6145 | Figure 8 1(a) shows a parallel plate capacitor C which
is a part of circuit through which a time-dependent
current i (t) flows Let us find the magnetic field at a
point such as P, in a region outside the parallel plate
capacitor For this, we consider a plane circular loop of
radius r whose plane is perpendicular to the direction
of the current-carrying wire, and which is centred
symmetrically with respect to the wire [Fig |
1 | 6143-6146 | 1(a) shows a parallel plate capacitor C which
is a part of circuit through which a time-dependent
current i (t) flows Let us find the magnetic field at a
point such as P, in a region outside the parallel plate
capacitor For this, we consider a plane circular loop of
radius r whose plane is perpendicular to the direction
of the current-carrying wire, and which is centred
symmetrically with respect to the wire [Fig 8 |
1 | 6144-6147 | Let us find the magnetic field at a
point such as P, in a region outside the parallel plate
capacitor For this, we consider a plane circular loop of
radius r whose plane is perpendicular to the direction
of the current-carrying wire, and which is centred
symmetrically with respect to the wire [Fig 8 1(a)] |
1 | 6145-6148 | For this, we consider a plane circular loop of
radius r whose plane is perpendicular to the direction
of the current-carrying wire, and which is centred
symmetrically with respect to the wire [Fig 8 1(a)] From
symmetry, the magnetic field is directed along the
circumference of the circular loop and is the same in
magnitude at all points on the loop so that if B is the
magnitude of the field, the left side of Eq |
1 | 6146-6149 | 8 1(a)] From
symmetry, the magnetic field is directed along the
circumference of the circular loop and is the same in
magnitude at all points on the loop so that if B is the
magnitude of the field, the left side of Eq (8 |
1 | 6147-6150 | 1(a)] From
symmetry, the magnetic field is directed along the
circumference of the circular loop and is the same in
magnitude at all points on the loop so that if B is the
magnitude of the field, the left side of Eq (8 1) is B (2p r) |
1 | 6148-6151 | From
symmetry, the magnetic field is directed along the
circumference of the circular loop and is the same in
magnitude at all points on the loop so that if B is the
magnitude of the field, the left side of Eq (8 1) is B (2p r) So we have
B (2pr) = m0i (t)
(8 |
1 | 6149-6152 | (8 1) is B (2p r) So we have
B (2pr) = m0i (t)
(8 2)
JAMES CLERK MAXWELL (1831–1879)
James Clerk Maxwell
(1831 – 1879) Born in
Edinburgh, Scotland,
was among the greatest
physicists
of
the
nineteenth century |
1 | 6150-6153 | 1) is B (2p r) So we have
B (2pr) = m0i (t)
(8 2)
JAMES CLERK MAXWELL (1831–1879)
James Clerk Maxwell
(1831 – 1879) Born in
Edinburgh, Scotland,
was among the greatest
physicists
of
the
nineteenth century He
derived the thermal
velocity distribution of
molecules in a gas and
was among the first to
obtain
reliable
estimates of molecular
parameters
from
measurable quantities
like
viscosity,
etc |
1 | 6151-6154 | So we have
B (2pr) = m0i (t)
(8 2)
JAMES CLERK MAXWELL (1831–1879)
James Clerk Maxwell
(1831 – 1879) Born in
Edinburgh, Scotland,
was among the greatest
physicists
of
the
nineteenth century He
derived the thermal
velocity distribution of
molecules in a gas and
was among the first to
obtain
reliable
estimates of molecular
parameters
from
measurable quantities
like
viscosity,
etc Maxwell’s
greatest
acheivement was the
unification of the laws of
electricity
and
magnetism (discovered
by Coulomb, Oersted,
Ampere and Faraday)
into a consistent set of
equations now called
Maxwell’s equations |
1 | 6152-6155 | 2)
JAMES CLERK MAXWELL (1831–1879)
James Clerk Maxwell
(1831 – 1879) Born in
Edinburgh, Scotland,
was among the greatest
physicists
of
the
nineteenth century He
derived the thermal
velocity distribution of
molecules in a gas and
was among the first to
obtain
reliable
estimates of molecular
parameters
from
measurable quantities
like
viscosity,
etc Maxwell’s
greatest
acheivement was the
unification of the laws of
electricity
and
magnetism (discovered
by Coulomb, Oersted,
Ampere and Faraday)
into a consistent set of
equations now called
Maxwell’s equations From these he arrived at
the most important
conclusion that light is
an
wave |
1 | 6153-6156 | He
derived the thermal
velocity distribution of
molecules in a gas and
was among the first to
obtain
reliable
estimates of molecular
parameters
from
measurable quantities
like
viscosity,
etc Maxwell’s
greatest
acheivement was the
unification of the laws of
electricity
and
magnetism (discovered
by Coulomb, Oersted,
Ampere and Faraday)
into a consistent set of
equations now called
Maxwell’s equations From these he arrived at
the most important
conclusion that light is
an
wave electromagnetic
Interestingly,
Maxwell did not agree
with the idea (strongly
suggested
by
the
Faraday’s
laws
of
electrolysis)
that
electricity
was
particulate in nature |
1 | 6154-6157 | Maxwell’s
greatest
acheivement was the
unification of the laws of
electricity
and
magnetism (discovered
by Coulomb, Oersted,
Ampere and Faraday)
into a consistent set of
equations now called
Maxwell’s equations From these he arrived at
the most important
conclusion that light is
an
wave electromagnetic
Interestingly,
Maxwell did not agree
with the idea (strongly
suggested
by
the
Faraday’s
laws
of
electrolysis)
that
electricity
was
particulate in nature Rationalised 2023-24
203
Electromagnetic
Waves
Now, consider a different surface, which has the same boundary |
1 | 6155-6158 | From these he arrived at
the most important
conclusion that light is
an
wave electromagnetic
Interestingly,
Maxwell did not agree
with the idea (strongly
suggested
by
the
Faraday’s
laws
of
electrolysis)
that
electricity
was
particulate in nature Rationalised 2023-24
203
Electromagnetic
Waves
Now, consider a different surface, which has the same boundary This
is a pot like surface [Fig |
1 | 6156-6159 | electromagnetic
Interestingly,
Maxwell did not agree
with the idea (strongly
suggested
by
the
Faraday’s
laws
of
electrolysis)
that
electricity
was
particulate in nature Rationalised 2023-24
203
Electromagnetic
Waves
Now, consider a different surface, which has the same boundary This
is a pot like surface [Fig 8 |
1 | 6157-6160 | Rationalised 2023-24
203
Electromagnetic
Waves
Now, consider a different surface, which has the same boundary This
is a pot like surface [Fig 8 1(b)] which nowhere touches the current, but
has its bottom between the capacitor plates; its mouth is the circular
loop mentioned above |
1 | 6158-6161 | This
is a pot like surface [Fig 8 1(b)] which nowhere touches the current, but
has its bottom between the capacitor plates; its mouth is the circular
loop mentioned above Another such surface is shaped like a tiffin box
(without the lid) [Fig |
1 | 6159-6162 | 8 1(b)] which nowhere touches the current, but
has its bottom between the capacitor plates; its mouth is the circular
loop mentioned above Another such surface is shaped like a tiffin box
(without the lid) [Fig 8 |
1 | 6160-6163 | 1(b)] which nowhere touches the current, but
has its bottom between the capacitor plates; its mouth is the circular
loop mentioned above Another such surface is shaped like a tiffin box
(without the lid) [Fig 8 1(c)] |
1 | 6161-6164 | Another such surface is shaped like a tiffin box
(without the lid) [Fig 8 1(c)] On applying Ampere’s circuital law to such
surfaces with the same perimeter, we find that the left hand side of
Eq |
1 | 6162-6165 | 8 1(c)] On applying Ampere’s circuital law to such
surfaces with the same perimeter, we find that the left hand side of
Eq (8 |
1 | 6163-6166 | 1(c)] On applying Ampere’s circuital law to such
surfaces with the same perimeter, we find that the left hand side of
Eq (8 1) has not changed but the right hand side is zero and not m0i,
since no current passes through the surface of Fig |
1 | 6164-6167 | On applying Ampere’s circuital law to such
surfaces with the same perimeter, we find that the left hand side of
Eq (8 1) has not changed but the right hand side is zero and not m0i,
since no current passes through the surface of Fig 8 |
1 | 6165-6168 | (8 1) has not changed but the right hand side is zero and not m0i,
since no current passes through the surface of Fig 8 1(b) and (c) |
1 | 6166-6169 | 1) has not changed but the right hand side is zero and not m0i,
since no current passes through the surface of Fig 8 1(b) and (c) So we
have a contradiction; calculated one way, there is a magnetic field at a
point P; calculated another way, the magnetic field at P is zero |
1 | 6167-6170 | 8 1(b) and (c) So we
have a contradiction; calculated one way, there is a magnetic field at a
point P; calculated another way, the magnetic field at P is zero Since the contradiction arises from our use of Ampere’s circuital law,
this law must be missing something |
1 | 6168-6171 | 1(b) and (c) So we
have a contradiction; calculated one way, there is a magnetic field at a
point P; calculated another way, the magnetic field at P is zero Since the contradiction arises from our use of Ampere’s circuital law,
this law must be missing something The missing term must be such
that one gets the same magnetic field at point P, no matter what surface
is used |
1 | 6169-6172 | So we
have a contradiction; calculated one way, there is a magnetic field at a
point P; calculated another way, the magnetic field at P is zero Since the contradiction arises from our use of Ampere’s circuital law,
this law must be missing something The missing term must be such
that one gets the same magnetic field at point P, no matter what surface
is used We can actually guess the missing term by looking carefully at
Fig |
1 | 6170-6173 | Since the contradiction arises from our use of Ampere’s circuital law,
this law must be missing something The missing term must be such
that one gets the same magnetic field at point P, no matter what surface
is used We can actually guess the missing term by looking carefully at
Fig 8 |
1 | 6171-6174 | The missing term must be such
that one gets the same magnetic field at point P, no matter what surface
is used We can actually guess the missing term by looking carefully at
Fig 8 1(c) |
1 | 6172-6175 | We can actually guess the missing term by looking carefully at
Fig 8 1(c) Is there anything passing through the surface S between the
plates of the capacitor |
1 | 6173-6176 | 8 1(c) Is there anything passing through the surface S between the
plates of the capacitor Yes, of course, the electric field |
1 | 6174-6177 | 1(c) Is there anything passing through the surface S between the
plates of the capacitor Yes, of course, the electric field If the plates of the
capacitor have an area A, and a total charge Q, the magnitude of the
electric field E between the plates is (Q/A)/e0 (see Eq |
1 | 6175-6178 | Is there anything passing through the surface S between the
plates of the capacitor Yes, of course, the electric field If the plates of the
capacitor have an area A, and a total charge Q, the magnitude of the
electric field E between the plates is (Q/A)/e0 (see Eq 2 |
1 | 6176-6179 | Yes, of course, the electric field If the plates of the
capacitor have an area A, and a total charge Q, the magnitude of the
electric field E between the plates is (Q/A)/e0 (see Eq 2 41) |
1 | 6177-6180 | If the plates of the
capacitor have an area A, and a total charge Q, the magnitude of the
electric field E between the plates is (Q/A)/e0 (see Eq 2 41) The field is
perpendicular to the surface S of Fig |
1 | 6178-6181 | 2 41) The field is
perpendicular to the surface S of Fig 8 |
1 | 6179-6182 | 41) The field is
perpendicular to the surface S of Fig 8 1(c) |
1 | 6180-6183 | The field is
perpendicular to the surface S of Fig 8 1(c) It has the same magnitude
over the area A of the capacitor plates, and vanishes outside it |
1 | 6181-6184 | 8 1(c) It has the same magnitude
over the area A of the capacitor plates, and vanishes outside it So what
is the electric flux FE through the surface S |
1 | 6182-6185 | 1(c) It has the same magnitude
over the area A of the capacitor plates, and vanishes outside it So what
is the electric flux FE through the surface S Using Gauss’s law, it is
E
0
0
1
=
=
Q
Q
A
AA
Φ
ε
=ε
E
(8 |
1 | 6183-6186 | It has the same magnitude
over the area A of the capacitor plates, and vanishes outside it So what
is the electric flux FE through the surface S Using Gauss’s law, it is
E
0
0
1
=
=
Q
Q
A
AA
Φ
ε
=ε
E
(8 3)
Now if the charge Q on the capacitor plates changes with time, there is a
current i = (dQ/dt), so that using Eq |
1 | 6184-6187 | So what
is the electric flux FE through the surface S Using Gauss’s law, it is
E
0
0
1
=
=
Q
Q
A
AA
Φ
ε
=ε
E
(8 3)
Now if the charge Q on the capacitor plates changes with time, there is a
current i = (dQ/dt), so that using Eq (8 |
1 | 6185-6188 | Using Gauss’s law, it is
E
0
0
1
=
=
Q
Q
A
AA
Φ
ε
=ε
E
(8 3)
Now if the charge Q on the capacitor plates changes with time, there is a
current i = (dQ/dt), so that using Eq (8 3), we have
d
d
dd
d
d
tΦE
t
Q
tQ
=
ε =
ε
0
0
1
This implies that for consistency,
ε0
d
d
tΦE
= i
(8 |
1 | 6186-6189 | 3)
Now if the charge Q on the capacitor plates changes with time, there is a
current i = (dQ/dt), so that using Eq (8 3), we have
d
d
dd
d
d
tΦE
t
Q
tQ
=
ε =
ε
0
0
1
This implies that for consistency,
ε0
d
d
tΦE
= i
(8 4)
This is the missing term in Ampere’s circuital law |
1 | 6187-6190 | (8 3), we have
d
d
dd
d
d
tΦE
t
Q
tQ
=
ε =
ε
0
0
1
This implies that for consistency,
ε0
d
d
tΦE
= i
(8 4)
This is the missing term in Ampere’s circuital law If we generalise
this law by adding to the total current carried by conductors through
the surface, another term which is e0 times the rate of change of electric
flux through the same surface, the total has the same value of current i
for all surfaces |
1 | 6188-6191 | 3), we have
d
d
dd
d
d
tΦE
t
Q
tQ
=
ε =
ε
0
0
1
This implies that for consistency,
ε0
d
d
tΦE
= i
(8 4)
This is the missing term in Ampere’s circuital law If we generalise
this law by adding to the total current carried by conductors through
the surface, another term which is e0 times the rate of change of electric
flux through the same surface, the total has the same value of current i
for all surfaces If this is done, there is no contradiction in the value of B
obtained anywhere using the generalised Ampere’s law |
1 | 6189-6192 | 4)
This is the missing term in Ampere’s circuital law If we generalise
this law by adding to the total current carried by conductors through
the surface, another term which is e0 times the rate of change of electric
flux through the same surface, the total has the same value of current i
for all surfaces If this is done, there is no contradiction in the value of B
obtained anywhere using the generalised Ampere’s law B at the point P
is non-zero no matter which surface is used for calculating it |
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