Chapter 31: Q. 64 (page 904)

Consider current $I$ passing through a resistor of radius $r$ , length $L$ , and resistance $R$ .
a. Determine the electric and magnetic fields at the surface of the resistor. Assume that the electric field is uniform throughout, including at the surface.
b. Determine the strength and direction of the Poynting vector at the surface of the resistor.
c. Show that the flux of the Poynting vector (i.e., the integral of $\vec{S} \cdot d \vec{A}$ ) over the surface of the resistor is $I^{2} R$ . Then give an interpretation of this result.

Expert verified

a) The electric and magnetic fields at the surface of the resistor, $∮ \cdot \vec{J} \cdot d \vec{A} = i \Rightarrow J = \frac{i}{π a^{2}}$

b) The strength and direction of the Poynting vector at the surface of the resistor, $E \cdot π a^{2} L = 4 π k Q enclosed$

c)The flux of the Poynting vector over the surface of the resistor, $Q = \frac{L^{2} i a ρ}{2 k}$

Step by step solution

The charge changes with respect to time is,

$\begin{aligned} q (t) = \int_{t_{0}}^{t_{1}} i d t \end{aligned}$

$\begin{aligned} ∮ \vec{J} \cdot d \vec{A} = i \Rightarrow J = \frac{i}{π a^{2}} \end{aligned}$

Charge enclosed is,

$Q_{enclosed} = J L$

Qenclosed=JLBy above equation, $B = \frac{μ_{0} I}{2 π a}$

Using ohms law,

$R = \frac{ρ L}{π a^{2}}$

Poynting vector shown as,

$\vec{S} = \frac{\vec{E}}{μ_{0}} \times \vec{B}$

By above equation, you got

$∮ \vec{E} d \vec{A} = 4 π k Q enclosed \Rightarrow E \cdot π a^{2} L = 4 π k Q_{enclosed}$ $(i = \frac{d q}{d t} .)$

Multiplication of electric field and magnetic field of resistors gives flux of the poynting vector,

$\begin{aligned} E B = \frac{i^{2} ρ L}{π a^{2}} \\ E = \frac{2 ρ L i}{μ_{0} a} \end{aligned}$

$Q = \frac{L^{2} i a ρ}{2 k}$

Unlock Step-by-Step Solutions & Ace Your Exams!

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.