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3690-3693
The field is directly proportional to the current and inversely proportional to the distance from the (infinitely long) current source (iv) There exists a simple rule to determine the direction of the magnetic field due to a long wire This rule, called the right-hand rule*, is: Grasp the wire in your right hand with your extended thumb pointing in the direction of the current Your fingers will curl around in the direction of the magnetic field
1
3691-3694
(iv) There exists a simple rule to determine the direction of the magnetic field due to a long wire This rule, called the right-hand rule*, is: Grasp the wire in your right hand with your extended thumb pointing in the direction of the current Your fingers will curl around in the direction of the magnetic field Ampere’s circuital law is not new in content from Biot-Savart law
1
3692-3695
This rule, called the right-hand rule*, is: Grasp the wire in your right hand with your extended thumb pointing in the direction of the current Your fingers will curl around in the direction of the magnetic field Ampere’s circuital law is not new in content from Biot-Savart law Both relate the magnetic field and the current, and both express the same physical consequences of a steady electrical current
1
3693-3696
Your fingers will curl around in the direction of the magnetic field Ampere’s circuital law is not new in content from Biot-Savart law Both relate the magnetic field and the current, and both express the same physical consequences of a steady electrical current Ampere’s law is to Biot-Savart law, what Gauss’s law is to Coulomb’s law
1
3694-3697
Ampere’s circuital law is not new in content from Biot-Savart law Both relate the magnetic field and the current, and both express the same physical consequences of a steady electrical current Ampere’s law is to Biot-Savart law, what Gauss’s law is to Coulomb’s law Both, Ampere’s and Gauss’s law relate a physical quantity on the periphery or boundary (magnetic or electric field) to another physical quantity, namely, the source, in the interior (current or charge)
1
3695-3698
Both relate the magnetic field and the current, and both express the same physical consequences of a steady electrical current Ampere’s law is to Biot-Savart law, what Gauss’s law is to Coulomb’s law Both, Ampere’s and Gauss’s law relate a physical quantity on the periphery or boundary (magnetic or electric field) to another physical quantity, namely, the source, in the interior (current or charge) We also note that Ampere’s circuital law holds for steady currents which do not fluctuate with time
1
3696-3699
Ampere’s law is to Biot-Savart law, what Gauss’s law is to Coulomb’s law Both, Ampere’s and Gauss’s law relate a physical quantity on the periphery or boundary (magnetic or electric field) to another physical quantity, namely, the source, in the interior (current or charge) We also note that Ampere’s circuital law holds for steady currents which do not fluctuate with time The following example will help us understand what is meant by the term enclosed current
1
3697-3700
Both, Ampere’s and Gauss’s law relate a physical quantity on the periphery or boundary (magnetic or electric field) to another physical quantity, namely, the source, in the interior (current or charge) We also note that Ampere’s circuital law holds for steady currents which do not fluctuate with time The following example will help us understand what is meant by the term enclosed current Example 4
1
3698-3701
We also note that Ampere’s circuital law holds for steady currents which do not fluctuate with time The following example will help us understand what is meant by the term enclosed current Example 4 8 Figure 4
1
3699-3702
The following example will help us understand what is meant by the term enclosed current Example 4 8 Figure 4 13 shows a long straight wire of a circular cross-section (radius a) carrying steady current I
1
3700-3703
Example 4 8 Figure 4 13 shows a long straight wire of a circular cross-section (radius a) carrying steady current I The current I is uniformly distributed across this cross-section
1
3701-3704
8 Figure 4 13 shows a long straight wire of a circular cross-section (radius a) carrying steady current I The current I is uniformly distributed across this cross-section Calculate the magnetic field in the region r < a and r > a
1
3702-3705
13 shows a long straight wire of a circular cross-section (radius a) carrying steady current I The current I is uniformly distributed across this cross-section Calculate the magnetic field in the region r < a and r > a FIGURE 4
1
3703-3706
The current I is uniformly distributed across this cross-section Calculate the magnetic field in the region r < a and r > a FIGURE 4 13 * Note that there are two distinct right-hand rules: One which gives the direction of B on the axis of current-loop and the other which gives direction of B for a straight conducting wire
1
3704-3707
Calculate the magnetic field in the region r < a and r > a FIGURE 4 13 * Note that there are two distinct right-hand rules: One which gives the direction of B on the axis of current-loop and the other which gives direction of B for a straight conducting wire Fingers and thumb play different roles in the two
1
3705-3708
FIGURE 4 13 * Note that there are two distinct right-hand rules: One which gives the direction of B on the axis of current-loop and the other which gives direction of B for a straight conducting wire Fingers and thumb play different roles in the two Rationalised 2023-24 Physics 120 EXAMPLE 4
1
3706-3709
13 * Note that there are two distinct right-hand rules: One which gives the direction of B on the axis of current-loop and the other which gives direction of B for a straight conducting wire Fingers and thumb play different roles in the two Rationalised 2023-24 Physics 120 EXAMPLE 4 8 Solution (a) Consider the case r > a
1
3707-3710
Fingers and thumb play different roles in the two Rationalised 2023-24 Physics 120 EXAMPLE 4 8 Solution (a) Consider the case r > a The Amperian loop, labelled 2, is a circle concentric with the cross-section
1
3708-3711
Rationalised 2023-24 Physics 120 EXAMPLE 4 8 Solution (a) Consider the case r > a The Amperian loop, labelled 2, is a circle concentric with the cross-section For this loop, L = 2 p r Ie = Current enclosed by the loop = I The result is the familiar expression for a long straight wire B (2p r) = m0I π 0 2 I B r =µ [4
1
3709-3712
8 Solution (a) Consider the case r > a The Amperian loop, labelled 2, is a circle concentric with the cross-section For this loop, L = 2 p r Ie = Current enclosed by the loop = I The result is the familiar expression for a long straight wire B (2p r) = m0I π 0 2 I B r =µ [4 19(a)] 1 B ∝r (r > a) Now the current enclosed Ie is not I, but is less than this value
1
3710-3713
The Amperian loop, labelled 2, is a circle concentric with the cross-section For this loop, L = 2 p r Ie = Current enclosed by the loop = I The result is the familiar expression for a long straight wire B (2p r) = m0I π 0 2 I B r =µ [4 19(a)] 1 B ∝r (r > a) Now the current enclosed Ie is not I, but is less than this value Since the current distribution is uniform, the current enclosed is, I I ar e =     π π 2 2 2 2 aIr = Using Ampere’s law, π 2 0 2 (2 ) I r B r a µ = B aI r =    µ0 2 2� [4
1
3711-3714
For this loop, L = 2 p r Ie = Current enclosed by the loop = I The result is the familiar expression for a long straight wire B (2p r) = m0I π 0 2 I B r =µ [4 19(a)] 1 B ∝r (r > a) Now the current enclosed Ie is not I, but is less than this value Since the current distribution is uniform, the current enclosed is, I I ar e =     π π 2 2 2 2 aIr = Using Ampere’s law, π 2 0 2 (2 ) I r B r a µ = B aI r =    µ0 2 2� [4 19(b)] B µ r (r < a) FIGURE 4
1
3712-3715
19(a)] 1 B ∝r (r > a) Now the current enclosed Ie is not I, but is less than this value Since the current distribution is uniform, the current enclosed is, I I ar e =     π π 2 2 2 2 aIr = Using Ampere’s law, π 2 0 2 (2 ) I r B r a µ = B aI r =    µ0 2 2� [4 19(b)] B µ r (r < a) FIGURE 4 14 Figure (4
1
3713-3716
Since the current distribution is uniform, the current enclosed is, I I ar e =     π π 2 2 2 2 aIr = Using Ampere’s law, π 2 0 2 (2 ) I r B r a µ = B aI r =    µ0 2 2� [4 19(b)] B µ r (r < a) FIGURE 4 14 Figure (4 14) shows a plot of the magnitude of B with distance r from the centre of the wire
1
3714-3717
19(b)] B µ r (r < a) FIGURE 4 14 Figure (4 14) shows a plot of the magnitude of B with distance r from the centre of the wire The direction of the field is tangential to the respective circular loop (1 or 2) and given by the right-hand rule described earlier in this section
1
3715-3718
14 Figure (4 14) shows a plot of the magnitude of B with distance r from the centre of the wire The direction of the field is tangential to the respective circular loop (1 or 2) and given by the right-hand rule described earlier in this section This example possesses the required symmetry so that Ampere’s law can be applied readily
1
3716-3719
14) shows a plot of the magnitude of B with distance r from the centre of the wire The direction of the field is tangential to the respective circular loop (1 or 2) and given by the right-hand rule described earlier in this section This example possesses the required symmetry so that Ampere’s law can be applied readily It should be noted that while Ampere’s circuital law holds for any loop, it may not always facilitate an evaluation of the magnetic field in every case
1
3717-3720
The direction of the field is tangential to the respective circular loop (1 or 2) and given by the right-hand rule described earlier in this section This example possesses the required symmetry so that Ampere’s law can be applied readily It should be noted that while Ampere’s circuital law holds for any loop, it may not always facilitate an evaluation of the magnetic field in every case For example, for the case of the circular loop discussed in Section 4
1
3718-3721
This example possesses the required symmetry so that Ampere’s law can be applied readily It should be noted that while Ampere’s circuital law holds for any loop, it may not always facilitate an evaluation of the magnetic field in every case For example, for the case of the circular loop discussed in Section 4 6, it cannot be applied to extract the simple expression B = m0I/2R [Eq
1
3719-3722
It should be noted that while Ampere’s circuital law holds for any loop, it may not always facilitate an evaluation of the magnetic field in every case For example, for the case of the circular loop discussed in Section 4 6, it cannot be applied to extract the simple expression B = m0I/2R [Eq (4
1
3720-3723
For example, for the case of the circular loop discussed in Section 4 6, it cannot be applied to extract the simple expression B = m0I/2R [Eq (4 16)] for the field at the centre of the loop
1
3721-3724
6, it cannot be applied to extract the simple expression B = m0I/2R [Eq (4 16)] for the field at the centre of the loop However, there exists a large number of situations of high symmetry where the law can be conveniently applied
1
3722-3725
(4 16)] for the field at the centre of the loop However, there exists a large number of situations of high symmetry where the law can be conveniently applied We shall use it in the next section to calculate p Rationalised 2023-24 121 Moving Charges and Magnetism the magnetic field produced by two commonly used and very useful magnetic systems: the solenoid and the toroid
1
3723-3726
16)] for the field at the centre of the loop However, there exists a large number of situations of high symmetry where the law can be conveniently applied We shall use it in the next section to calculate p Rationalised 2023-24 121 Moving Charges and Magnetism the magnetic field produced by two commonly used and very useful magnetic systems: the solenoid and the toroid 4
1
3724-3727
However, there exists a large number of situations of high symmetry where the law can be conveniently applied We shall use it in the next section to calculate p Rationalised 2023-24 121 Moving Charges and Magnetism the magnetic field produced by two commonly used and very useful magnetic systems: the solenoid and the toroid 4 7 THE SOLENOID We shall discuss a long solenoid
1
3725-3728
We shall use it in the next section to calculate p Rationalised 2023-24 121 Moving Charges and Magnetism the magnetic field produced by two commonly used and very useful magnetic systems: the solenoid and the toroid 4 7 THE SOLENOID We shall discuss a long solenoid By long solenoid we mean that the solenoid’s length is large compared to its radius
1
3726-3729
4 7 THE SOLENOID We shall discuss a long solenoid By long solenoid we mean that the solenoid’s length is large compared to its radius It consists of a long wire wound in the form of a helix where the neighbouring turns are closely spaced
1
3727-3730
7 THE SOLENOID We shall discuss a long solenoid By long solenoid we mean that the solenoid’s length is large compared to its radius It consists of a long wire wound in the form of a helix where the neighbouring turns are closely spaced So each turn can be regarded as a circular loop
1
3728-3731
By long solenoid we mean that the solenoid’s length is large compared to its radius It consists of a long wire wound in the form of a helix where the neighbouring turns are closely spaced So each turn can be regarded as a circular loop The net magnetic field is the vector sum of the fields due to all the turns
1
3729-3732
It consists of a long wire wound in the form of a helix where the neighbouring turns are closely spaced So each turn can be regarded as a circular loop The net magnetic field is the vector sum of the fields due to all the turns Enamelled wires are used for winding so that turns are insulated from each other
1
3730-3733
So each turn can be regarded as a circular loop The net magnetic field is the vector sum of the fields due to all the turns Enamelled wires are used for winding so that turns are insulated from each other Figure 4
1
3731-3734
The net magnetic field is the vector sum of the fields due to all the turns Enamelled wires are used for winding so that turns are insulated from each other Figure 4 15 displays the magnetic field lines for a finite solenoid
1
3732-3735
Enamelled wires are used for winding so that turns are insulated from each other Figure 4 15 displays the magnetic field lines for a finite solenoid We show a section of this solenoid in an enlarged manner in Fig
1
3733-3736
Figure 4 15 displays the magnetic field lines for a finite solenoid We show a section of this solenoid in an enlarged manner in Fig 4
1
3734-3737
15 displays the magnetic field lines for a finite solenoid We show a section of this solenoid in an enlarged manner in Fig 4 15(a)
1
3735-3738
We show a section of this solenoid in an enlarged manner in Fig 4 15(a) Figure 4
1
3736-3739
4 15(a) Figure 4 15(b) shows the entire finite solenoid with its magnetic field
1
3737-3740
15(a) Figure 4 15(b) shows the entire finite solenoid with its magnetic field In Fig
1
3738-3741
Figure 4 15(b) shows the entire finite solenoid with its magnetic field In Fig 4
1
3739-3742
15(b) shows the entire finite solenoid with its magnetic field In Fig 4 15(a), it is clear from the circular loops that the field between two neighbouring turns vanishes
1
3740-3743
In Fig 4 15(a), it is clear from the circular loops that the field between two neighbouring turns vanishes In Fig
1
3741-3744
4 15(a), it is clear from the circular loops that the field between two neighbouring turns vanishes In Fig 4
1
3742-3745
15(a), it is clear from the circular loops that the field between two neighbouring turns vanishes In Fig 4 15(b), we see that the field at the interior mid-point P is uniform, strong and along the axis of the solenoid
1
3743-3746
In Fig 4 15(b), we see that the field at the interior mid-point P is uniform, strong and along the axis of the solenoid The field at the exterior mid-point Q is weak and moreover is along the axis of the solenoid with no perpendicular or normal component
1
3744-3747
4 15(b), we see that the field at the interior mid-point P is uniform, strong and along the axis of the solenoid The field at the exterior mid-point Q is weak and moreover is along the axis of the solenoid with no perpendicular or normal component As the FIGURE 4
1
3745-3748
15(b), we see that the field at the interior mid-point P is uniform, strong and along the axis of the solenoid The field at the exterior mid-point Q is weak and moreover is along the axis of the solenoid with no perpendicular or normal component As the FIGURE 4 15 (a) The magnetic field due to a section of the solenoid which has been stretched out for clarity
1
3746-3749
The field at the exterior mid-point Q is weak and moreover is along the axis of the solenoid with no perpendicular or normal component As the FIGURE 4 15 (a) The magnetic field due to a section of the solenoid which has been stretched out for clarity Only the exterior semi-circular part is shown
1
3747-3750
As the FIGURE 4 15 (a) The magnetic field due to a section of the solenoid which has been stretched out for clarity Only the exterior semi-circular part is shown Notice how the circular loops between neighbouring turns tend to cancel
1
3748-3751
15 (a) The magnetic field due to a section of the solenoid which has been stretched out for clarity Only the exterior semi-circular part is shown Notice how the circular loops between neighbouring turns tend to cancel (b) The magnetic field of a finite solenoid
1
3749-3752
Only the exterior semi-circular part is shown Notice how the circular loops between neighbouring turns tend to cancel (b) The magnetic field of a finite solenoid FIGURE 4
1
3750-3753
Notice how the circular loops between neighbouring turns tend to cancel (b) The magnetic field of a finite solenoid FIGURE 4 16 The magnetic field of a very long solenoid
1
3751-3754
(b) The magnetic field of a finite solenoid FIGURE 4 16 The magnetic field of a very long solenoid We consider a rectangular Amperian loop abcd to determine the field
1
3752-3755
FIGURE 4 16 The magnetic field of a very long solenoid We consider a rectangular Amperian loop abcd to determine the field Rationalised 2023-24 Physics 122 EXAMPLE 4
1
3753-3756
16 The magnetic field of a very long solenoid We consider a rectangular Amperian loop abcd to determine the field Rationalised 2023-24 Physics 122 EXAMPLE 4 9 solenoid is made longer it appears like a long cylindrical metal sheet
1
3754-3757
We consider a rectangular Amperian loop abcd to determine the field Rationalised 2023-24 Physics 122 EXAMPLE 4 9 solenoid is made longer it appears like a long cylindrical metal sheet Figure 4
1
3755-3758
Rationalised 2023-24 Physics 122 EXAMPLE 4 9 solenoid is made longer it appears like a long cylindrical metal sheet Figure 4 16 represents this idealised picture
1
3756-3759
9 solenoid is made longer it appears like a long cylindrical metal sheet Figure 4 16 represents this idealised picture The field outside the solenoid approaches zero
1
3757-3760
Figure 4 16 represents this idealised picture The field outside the solenoid approaches zero We shall assume that the field outside is zero
1
3758-3761
16 represents this idealised picture The field outside the solenoid approaches zero We shall assume that the field outside is zero The field inside becomes everywhere parallel to the axis
1
3759-3762
The field outside the solenoid approaches zero We shall assume that the field outside is zero The field inside becomes everywhere parallel to the axis Consider a rectangular Amperian loop abcd
1
3760-3763
We shall assume that the field outside is zero The field inside becomes everywhere parallel to the axis Consider a rectangular Amperian loop abcd Along cd the field is zero as argued above
1
3761-3764
The field inside becomes everywhere parallel to the axis Consider a rectangular Amperian loop abcd Along cd the field is zero as argued above Along transverse sections bc and ad, the field component is zero
1
3762-3765
Consider a rectangular Amperian loop abcd Along cd the field is zero as argued above Along transverse sections bc and ad, the field component is zero Thus, these two sections make no contribution
1
3763-3766
Along cd the field is zero as argued above Along transverse sections bc and ad, the field component is zero Thus, these two sections make no contribution Let the field along ab be B
1
3764-3767
Along transverse sections bc and ad, the field component is zero Thus, these two sections make no contribution Let the field along ab be B Thus, the relevant length of the Amperian loop is, L = h
1
3765-3768
Thus, these two sections make no contribution Let the field along ab be B Thus, the relevant length of the Amperian loop is, L = h Let n be the number of turns per unit length, then the total number of turns is nh
1
3766-3769
Let the field along ab be B Thus, the relevant length of the Amperian loop is, L = h Let n be the number of turns per unit length, then the total number of turns is nh The enclosed current is, Ie = I (n h), where I is the current in the solenoid
1
3767-3770
Thus, the relevant length of the Amperian loop is, L = h Let n be the number of turns per unit length, then the total number of turns is nh The enclosed current is, Ie = I (n h), where I is the current in the solenoid From Ampere’s circuital law [Eq
1
3768-3771
Let n be the number of turns per unit length, then the total number of turns is nh The enclosed current is, Ie = I (n h), where I is the current in the solenoid From Ampere’s circuital law [Eq 4
1
3769-3772
The enclosed current is, Ie = I (n h), where I is the current in the solenoid From Ampere’s circuital law [Eq 4 17 (b)] BL = m0Ie, B h = m0I (n h) B = m0 n I (4
1
3770-3773
From Ampere’s circuital law [Eq 4 17 (b)] BL = m0Ie, B h = m0I (n h) B = m0 n I (4 20) The direction of the field is given by the right-hand rule
1
3771-3774
4 17 (b)] BL = m0Ie, B h = m0I (n h) B = m0 n I (4 20) The direction of the field is given by the right-hand rule The solenoid is commonly used to obtain a uniform magnetic field
1
3772-3775
17 (b)] BL = m0Ie, B h = m0I (n h) B = m0 n I (4 20) The direction of the field is given by the right-hand rule The solenoid is commonly used to obtain a uniform magnetic field We shall see in the next chapter that a large field is possible by inserting a soft iron core inside the solenoid
1
3773-3776
20) The direction of the field is given by the right-hand rule The solenoid is commonly used to obtain a uniform magnetic field We shall see in the next chapter that a large field is possible by inserting a soft iron core inside the solenoid Example 4
1
3774-3777
The solenoid is commonly used to obtain a uniform magnetic field We shall see in the next chapter that a large field is possible by inserting a soft iron core inside the solenoid Example 4 9 A solenoid of length 0
1
3775-3778
We shall see in the next chapter that a large field is possible by inserting a soft iron core inside the solenoid Example 4 9 A solenoid of length 0 5 m has a radius of 1 cm and is made up of 500 turns
1
3776-3779
Example 4 9 A solenoid of length 0 5 m has a radius of 1 cm and is made up of 500 turns It carries a current of 5 A
1
3777-3780
9 A solenoid of length 0 5 m has a radius of 1 cm and is made up of 500 turns It carries a current of 5 A What is the magnitude of the magnetic field inside the solenoid
1
3778-3781
5 m has a radius of 1 cm and is made up of 500 turns It carries a current of 5 A What is the magnitude of the magnetic field inside the solenoid Solution The number of turns per unit length is, 500 1000 n =0
1
3779-3782
It carries a current of 5 A What is the magnitude of the magnetic field inside the solenoid Solution The number of turns per unit length is, 500 1000 n =0 5 = turns/m The length l = 0
1
3780-3783
What is the magnitude of the magnetic field inside the solenoid Solution The number of turns per unit length is, 500 1000 n =0 5 = turns/m The length l = 0 5 m and radius r = 0
1
3781-3784
Solution The number of turns per unit length is, 500 1000 n =0 5 = turns/m The length l = 0 5 m and radius r = 0 01 m
1
3782-3785
5 = turns/m The length l = 0 5 m and radius r = 0 01 m Thus, l/a = 50 i
1
3783-3786
5 m and radius r = 0 01 m Thus, l/a = 50 i e
1
3784-3787
01 m Thus, l/a = 50 i e , l >> a
1
3785-3788
Thus, l/a = 50 i e , l >> a Hence, we can use the long solenoid formula, namely, Eq
1
3786-3789
e , l >> a Hence, we can use the long solenoid formula, namely, Eq (4
1
3787-3790
, l >> a Hence, we can use the long solenoid formula, namely, Eq (4 20) B = m0n I = 4p × 10–7 × 103 × 5 = 6
1
3788-3791
Hence, we can use the long solenoid formula, namely, Eq (4 20) B = m0n I = 4p × 10–7 × 103 × 5 = 6 28 × 10–3 T FIGURE 4
1
3789-3792
(4 20) B = m0n I = 4p × 10–7 × 103 × 5 = 6 28 × 10–3 T FIGURE 4 17 Two long straight parallel conductors carrying steady currents Ia and Ib and separated by a distance d