A conducting hollow cylinder (inner radius = a, outer radius = b) carries a current of 40 A that is uniformly distributed over the cross section of the conductor. If a = 3.0 mm and b = 6.0 mm, what is the magnitude of the (line) integral around a circular path (radius = 5.0 mm) centered on the axis of the cylinder and in a plane perpendicular to that axis?
A) 50 μT ⋅ m
B) 30 μT ⋅ m
C) 22 μT ⋅ m
D) 37 μT ⋅ m
E) 47 μT ⋅ m

Question

A conducting hollow cylinder (inner radius = a, outer radius = b) carries a current of 40 A that is uniformly distributed over the cross section of the conductor. If a = 3.0 mm and b = 6.0 mm, what is the magnitude of the (line) integral  around a circular path (radius = 5.0 mm) centered on the axis of the cylinder and in a plane perpendicular to that axis?
A) 50 μT ⋅ m 
B) 30 μT ⋅ m 
C) 22 μT ⋅ m 
D) 37 μT ⋅ m 
E) 47 μT ⋅ m

Quizplus · Accepted Answer

We can use Ampere's Law to solve this problem. First, we need to determine the total current enclosed by the circular path. Since the current is uniformly distributed over the cross section of the conductor, the cross-sectional area of the conductor is $\pi(b^2-a^2)$, and the total current is:

$I = 	ext{current density} 	imes 	ext{cross-sectional area} = \frac{40}{\pi(b^2-a^2)}\pi(b^2-a^2) = 40$

Now we can use Ampere's Law:

$\oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{enc}$

where $I_{enc}$ is the total current enclosed by the circular path, and $\mu_0$ is the permeability of free space.

We can use the symmetry of the problem to determine that the magnetic field $\mathbf{B}$ encircles the current carrying cylinder with concentric circles, and is perpendicular to the circular path of integration. Therefore, we can simplify the line integral to:

$\oint \mathbf{B} \cdot d\mathbf{l} = B\oint dl = Bl$

where $l$ is the length of the circular path of integration.

Solving for $B$, we get:

$B = \frac{1}{l} \oint \mathbf{B} \cdot d\mathbf{l} = \frac{\mu_0 I_{enc}}{2\pi l}$

Plugging in the values we have determined, we get:

$B = \frac{\mu_0 	imes 40}{2\pi 	imes 0.005} = 31.8 \, \mu T$

which is closest to answer choice B.

A Conducting Hollow Cylinder (Inner Radius = A, Outer Radius