A solid cylindrical conducting shell of inner radius a = 4.8 cm and outer radius

Question

A solid cylindrical conducting shell of inner radius a = 4.8 cm and outer radius b = 5.9 cm has its axis aligned with the z-axis as shown. It carries a uniformly distributed current I2= 7.2 A in the positive z-direction. An inifinte conducting wire is located along the z-axis and carries a current I1 = 2.7 A in the negative z-direction.

1)

What is By(P), the y-component of the magnetic field at point P, located a distance d = 21 cm from the origin along the x-axis as shown?

___________T

2)

What is

where the integral is taken along the dotted path shown in the figure above: first from point P to point R at (x,y) = (0.707d, 0.707d), and then to point S at (x,y) = (0.6d, 0.6d).

________T-m

3)

What is By(T), the y-component of the magnetic field at point T, located at (x,y) = (-5.2 cm,0), as shown?

__________T

4)

What is

where the integral is taken on the straight line path from point S to point P as shown?

________T-m

5)

Suppose the magnitude of the current I2 is now doubled. How does the magnitude of the magnetic field at (x,y) = (2.4 cm, 0) change?

B(2.4 cm, 0) increases

B(2.4 cm, 0) decreases

B(2.4 cm, 0) remains the same

Explanation / Answer

1)The magnetic field at P will be given by:

Int(B.dl) = mu0 I(enclosed)

B(2 pi d) = mu (I1 - I2)

B = mu0 (I1 - I2)/2 pi d

B = 4 pi x 10^-7 x (2.7 - 7.2)/2 pi x 0.21 = -4.286 x 10^-6 T

By = - 4.286 x 10^-6 T

2)Int (B.dl) = 45/360 mu0 (I1 - I2)

Int (B.dl) = 1/8 x 4 pi x 10^-7 x (2.7 - 7.2) = - 7.07 x 10^-7 T-m

3)Bt = mu0 Ienc/ 2 pi T

By(t) = mu0/ 2 pi T [I1 - I2(T^2 - a^2/b^2 - a^2)]

By(t) = 4 pi x 10^-7 / 2 pi x 0.052 [ 2.7 - 7.2 (0.052^2 - 0.048^2/0.059^2 - 0.048^2)] = 9.73 x 10^-7 T

4)2)Int (B.dl) = 45/360 mu0 (I1 - I2)

Int (B.dl) = 1/8 x 4 pi x 10^-7 x (2.7 - 7.2) = + 7.07 x 10^-7 T-m

5)B(2.4 cm, 0) remains the same

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