Coaxial cables are often used to connect electrical devices, such as your stereo system, and in receiving signals in TV cable systems. Model a long coaxial cable as consisting of two thin concentric cylindrical conducting shells of radii a and b and length l, as in Figure 32.13. The

Question

Coaxial cables are often used to connect electrical devices, such as your stereo system, and in receiving signals in TV cable systems. Model a long coaxial cable as consisting of two thin concentric cylindrical conducting shells of radii a and b and length $\ell,$ , as in Figure 32.13. The conducting shells carry the same current I in opposite directions. Imagine that the inner conductor carries current to a device and that the outer one acts as a return path carrying the current back to the source.

(A) Calculate the self-inductance L of this cable.

(B) Calculate the total energy stored in the magnetic field of the cable.

(A) Calculate the self-inductance L of this cable.

(B) Calculate the total energy stored in the magnetic field of the cable.

Accepted Answer

Conceptualize the situation with the help of Figure 32.13. While we do not have a visible coil in this geometry, imagine a thin radial slice of the coaxial cable, such as the light blue rectangle in Figure 32.13. If we consider the inner and outer conductors to be connected at the ends of the cable (above and below the figure), this slice represents one large conducting loop. The current in the loop sets up a magnetic field between the inner and outer conductors that passes through this loop. If the current changes, the magnetic field changes and the induced emf opposes the original change in the current in the conductors. We categorize this situation as one in which we can calculate an inductance, but we must return to the fundamental definition of inductance, Equation 32.2. To analyze the problem and obtain L, we must find the magnetic flux through the light blue rectangle in Figure 32.13. Ampère’s law (see Section 30.3) tells us that the magnetic field in the region between the shells is due to the inner conductor and its magnitude is [latex] B=\mu_{0} I / 2 \pi r [/latex] where r is measured from the common center of the shells. The magnetic field is zero outside the outer shell (r > b) because the net current passing through the area enclosed by a circular path surrounding the cable is zero, and hence from Ampère’s law [latex] \oint B \cdot d s =0 [/latex] The magnetic field is zero inside the inner shell because the shell is hollow and no current is present within a radius r < a.

The magnetic field is perpendicular to the light blue rectangle of length [latex] \ell [/latex] and width b - a, the cross section of interest. Because the magnetic field varies with radial position across this rectangle, we must use calculus to find the total magnetic flux. Dividing this rectangle into strips of width dr, such as the dark blue strip in Figure 32.13, we see that the area of each strip is [latex] \ell [/latex]dr and that the flux through each strip is [latex] B d A=B \ell d r [/latex]. Hence, we find the total flux through the entire cross section by integrating:

[latex] U=\frac{1}{2} L I^{2} [/latex] (32.12).

[latex] \Phi_{B}=\int B d A=\int_{a}^{b} \frac{\mu_{0} I}{2 \pi r} \ell d r=\frac{\mu_{0} I \ell}{2 \pi} \int_{a}^{b} \frac{d r}{r}=\frac{\mu_{0} I \ell}{2 \pi} \ln \left(\frac{b}{a}\right) [/latex].

Using this result, we find that the self-inductance of the cable is

[latex] L=\frac{\Phi_{B}}{I}=\frac{\mu_{0} \ell}{2 \pi} \ln \left(\frac{b}{a}\right) [/latex].

(B) Calculate the total energy stored in the magnetic field of the cable.

Using Equation 32.12 and the results to part (A) gives

[latex] U=\frac{1}{2} L I^{2} [/latex] (32.12).

[latex] U=\frac{1}{2} L I^{2}=\frac{\mu_{0} \ell I^{2}}{4 \pi} \ln \left(\frac{b}{a}\right) [/latex].

To finalize the problem, note that the inductance increases if [latex] \ell[/latex] increases, if b increases, or if a decreases. This is consistent with our conceptualization—any of these changes increases the size of the loop represented by our radial slice and through which the magnetic field passes; this increases the inductance.

Question 32.5: Coaxial cables are often used to connect electrical devices,...

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