Magnetic Fields from Currents in a Wire and a Cylindrical Shell 1 2 3 4 5 6 A so

Question

Magnetic Fields from Currents in a Wire and a Cylindrical Shell

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A solid cylindrical conducting shell of inner radius a = 4.5 cm and outer radius b = 6.4 cm has its axis aligned with the z-axis as shown. It carries a uniformly distributed current I2= 6.1 A in the positive z-direction. An inifinte conducting wire is located along the z-axis and carries a current I1 = 3.2 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 = 23 cm from the origin along the x-axis as shown?

T

2)

What is

PSB dl PSBdl

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) = (-4.9 cm,0), as shown?

T

4)

What is

SPB dl SPBdl

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

T-m

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 (3.2 - 6.1)/2 pi x 0.23 = - 2.52 x 10^-6 T

By = - 2.52 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 (3.2 - 6.1) = - 4.55 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 0.049 [ 3.2 - 6.1 (0.053^2 - 0.048^2/0.069^2 - 0.048^2)] = 8 x 10^-6 T

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

Int (B.dl) = 1/8 x 4 pi x 10^-7 x (6.1 - 3.2) = + 4.55 x 10^-7 T-m

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