Question

If a long hollow copper pipe carries a direct current, the magnetic field associated with the current will be (a) Only inside the pipe (b) Only outside the pipe (c) Neither inside nor outside the pipe (d) Both inside and outside the pipe

Short answer

Using Ampere's law, we found that the magnetic field is zero inside the hollow copper pipe when the radius is less than the pipe's inner radius and nonzero outside the pipe when the radius is greater than the pipe's outer radius. Therefore, the correct answer is (b) Only outside the pipe.

Step-by-step solution

Recall Ampere's Law

Ampere's law states that the closed-line integral of the magnetic field (B) around a closed loop is equal to the product of the permeability constant (μ₀) and the total current (I) passing through the loop. Mathematically, the law can be represented as follows: \[\oint_{loop} \vec{B} \cdot d\vec{l} = \mu_0 I\] Keep in mind that this law will be useful to analyze the magnetic field distribution for the given hollow copper pipe.

Consider a long hollow copper pipe

Let's consider a long hollow copper pipe with direct current, I, flowing through it. In this problem, we need to determine at which region the magnetic field is present (inside the pipe, outside the pipe, or both).

Analyze the magnetic field inside and outside the pipe

For understanding the distribution of the magnetic field, we can choose two Amperian loops: (a) An Amperian loop inside the Copper pipe (radius < pipe's inner radius): Since there is no current passing through this loop, according to Ampere's law, the magnetic field inside the pipe (when radius < pipe's inner radius) will be zero. (b) An Amperian loop outside the Copper pipe (radius > pipe's outer radius): Now, the total current (I) is enclosed by this loop. Applying Ampere's law to this loop, we find that the magnetic field outside the pipe (when radius > pipe's outer radius) is nonzero. From the analysis above, we can conclude the distribution of the magnetic field associated with the direct current flowing through the long hollow copper pipe.

Choose the correct option

After analyzing the magnetic field using Ampere's law, we can now choose the correct option: (a) Only inside the pipe -> This option is incorrect, as we found that the magnetic field is zero inside the pipe. (b) Only outside the pipe -> This option is correct, as we found that the magnetic field is nonzero outside the pipe. (c) Neither inside nor outside the pipe -> This option is incorrect, as we found that there is a magnetic field outside the pipe. (d) Both inside and outside the pipe -> This option is incorrect, as we found that the magnetic field is zero inside the pipe but nonzero outside the pipe. So, the correct answer is (b) Only outside the pipe.