Question

A length of copper wire carries a current of 10 A uniformly distributed through its cross section. (a) Calculate the energy density of the magnetic field. (b) Calculate the energy density of the electric field at the surface of the wire. The wire diameter is 2.5 mm, and its resistance per unit length is$\mathbf{3}\mathbf{.}\mathbf{3}\mathbf{\Omega }\mathbf{/}\mathbf{km}$.

Short answer

a) Energy density of the magnetic field at the surface of the wire is $1.0\mathrm{J}/{\mathrm{m}}^3$.

b) Energy density of the electric field at the surface of the wire is $4.8\times {10}^{-15}\mathrm{J}/{\mathrm{m}}^3$.

Step-by-step solution

Given

$$\begin{gathered} I=10\mathrm{A} \\ d=2.5\mathrm{mm}=2.5\times {10}^{-3}\mathrm{m} \end{gathered}$$

Resistance per unit length$\frac{R}{I}=3.3\frac{\Omega }{\mathrm{km}}=3.3\times {10}^{-3}\Omega /\mathrm{m}$

Understanding the concept

We need to use the formula for the energy density for the magnetic field and the electric field at the surface of the wire. Using these equations and the given data, we can calculate the energy density for both, the electric field and the magnetic field.

Formula:

$$\begin{gathered} u_B=\frac{B^2}{2{\mu }_0} \\ B=\frac{{\mu }_{0}I}{2R} \\ {u}_E=\frac{1}{2}\epsilon E^2 \end{gathered}$$

Calculate the energy density of the magnetic field at the surface of the wire .

We have,

$$\begin{gathered} u_B=\frac{B^2}{2{\mu }_0} \\ \mathrm{But}, \\ \mathrm{B}=\frac{{\mathrm{\mu }}_0\mathrm{I}}{2\mathrm{R}} \\ \mathrm{So} \\ {\mathrm{u}}_{\mathrm{B}}=\frac{1}{2{\mathrm{\mu }}_0}\times \left(\frac{{\mathrm{\mu }}_0\mathrm{I}}{2\mathrm{R}}\right) \\ {\mathrm{u}}_{\mathrm{B}}=\frac{{\mathrm{\mu }}_0{\mathrm{I}}^2}{2{\mathrm{R}}^2} \\ {\mathrm{u}}_{\mathrm{B}}=\frac{4\mathrm{\pi }\times {10}^{-7}\times {10}^2}{8\times \left({\displaystyle \frac{2.5\times {10}^{-3}}{2}}\right)} \\ {\mathrm{u}}_{\mathrm{B}}=1.0\mathrm{J}/{\mathrm{m}}^3 \end{gathered}$$

(b) Calculate Energy density of the electric field at the surface of the wire.

We have,

$$\begin{gathered} u_E=\frac{1}{2}\epsilon E^2 \\ \mathrm{But}, \\ \mathrm{E}=\mathrm{\rho J} \\ {u}_E=\frac{1}{2}\epsilon {\left(\mathrm{\rho J}\right)}^2 \\ \mathrm{But}, \\ \mathrm{J}=\mathrm{I}/\mathrm{A} \\ {u}_E=\frac{1}{2}\epsilon {\left(\frac{\rho I}{A}\right)}^2 \end{gathered}$$

We multiply and divide inside the bracket by

${\mathrm{u}}_{\mathrm{E}}=\frac{1}{2}\mathrm{\epsilon }{\left(\frac{\rho II}{AI}\right)}^2$

But, we know that,

$$\begin{gathered} R=\frac{\rho I}{A} \\ {u}_E=\frac{\epsilon }{2}{\left(\frac{RI}{I}\right)}^2 \\ {u}_E=\frac{8.85\times {10}^{-12}}{2}\times {\left(10\times 3.3\times {10}^{-3}\right)}^2 \\ {u}_E=4.8\times {10}^{-15}\mathrm{J}/{\mathrm{m}}^3 \end{gathered}$$