Suppose that we have two separate infinite cylindrically symmetric pipes. One has an inner radius of $R_{1}$, an outer radius of $R_{2}$, and carries in its hull a uniform current density $J_{0}$ parallel to its central axis. The other has an inner radius of $2 R_{1}$, an outer radius of $2 R_{2}$, and carries the same current density (magnitude and direction) in its hull. How does the magnitude of the magnetic field $\left|B_{2}{ }_{2}\right|$ just outside the pipe with the larger radius compare to the field magnitude $\left|B^{\overrightarrow{ }}{ }_{1}\right|$ just outside the smaller pipe?
A) $\left|\vec{B}_{2}\right|=4\left|\overrightarrow{B_{1}}\right|$
B) $\left|\vec{B}_{2}\right|=2\left|\vec{B}_{1}\right|$
C) $\left|B_{2}\right|=\left|B^{\prime}\right|$
D) $\left|B_{2}\right|=1 / 2\left|B_{1}\right|$
E) $\left|\overrightarrow{B_{2}}\right|=1 / 4\left|\overrightarrow{B_{1}}\right|$
F) Some other ratio (specify)

Question

Quizplus · Accepted Answer

The magnetic field just outside a cylindrical pipe carrying a uniform current is directly proportional to the total current flowing through the pipe and inversely proportional to the distance from the center of the pipe. Since both pipes carry the same current density and the second pipe has twice the dimensions of the first, the total current in both pipes is the same. However, the magnetic field's magnitude at a point just outside each pipe depends on the distance from the center (which is proportional to the radius of the pipe). Since the radius of the second pipe is twice that of the first, the magnetic field just outside the second pipe ($\left|\vec{B}_{2}\right|$) will be the same as that just outside the first pipe ($\left|\vec{B}_{1}\right|$), not twice, making my initial explanation incorrect in suggesting a direct relationship based on radius alone for the magnetic field magnitude just outside the pipes. The correct approach involves considering the total current enclosed by a path following Ampère's Law, which would indeed show that the magnetic fields just outside both pipes are equal if the current densities and total currents are the same, leading to a correction in understanding that the magnitudes of the magnetic fields just outside both pipes would be equal, not different as previously stated.

Suppose That We Have Two Separate Infinite Cylindrically Symmetric Pipes R1R_{1}R1​

Suppose That We Have Two Separate Infinite Cylindrically Symmetric Pipes $R_{1}$