Electromagnetic Fields and Waves: Notes, Solved Problems and Interactive Lab

Exercise 4.2.4

About the further exercises of Section 4.2

Ten further exercises in the spirit of Problems 4.2.1–4.2.5, the worked examples and the home exercises, graded from easy to harder. Exercises 4.2.1–4.2.3 are about magnetic media and the boundary conditions, Exercises 4.2.4–4.2.5 about Ohm's law and the Joule phenomenon, Exercises 4.2.6–4.2.8 about inductance and energy, and Exercises 4.2.9–4.2.10 about dipoles and coupled circuits. They use only

B = μ_{0} H + M = μ H, J_{m} = \nabla \times \frac{M}{μ _{0}}, J_{m s} = \frac{1}{μ _{0}} M \times a_{n}, H_{1 τ} - H_{2 τ} = J_{n S}, B_{1 n} = B_{2 n}, L_{12} = \frac{N _{2} Φ _{12}}{I _{1}}, W_{m} = \frac{1}{2} \int H \cdot B d V = \frac{1}{2} L I^{2}, J = σ E, p = J \cdot E, T = m \times B .

As in the lectures, $M$ carries the units of $B$ (tesla), so $M / μ_{0}$ is the magnetisation in A/m.

Exercise 4.2.4

A heater wire

cf. Ohm's law and the Joule phenomenon

A heating element is a nichrome wire ( $σ = 9.1 \times 1 0^{5} S/m$ ) of length $l = 2 m$ and diameter $0.5 mm$ connected to $V = 230 V$ . Find (a) the resistance and the current, (b) $J$ and $E$ inside the wire, (c) the power density $p = J \cdot E$ and the total power, checked against $V I$ . (d) Copper has $σ = 5.8 \times 1 0^{7} S/m$ : what length of the same copper wire would dissipate the same power at $230 V$ , and why is it not used?

Try it yourself first.

Exercise 4.2.4: A heater wire