Electricity and Magnetism · Problem 9.6

ELECTRIC FIELDS IN MATTER — Edward Purcell & David Morin

Physics — E&M · em-waves-solenoid

Problem

A solenoid with radius RR has nn turns per unit length. The current varies with time according to I(t)=I0cos⁡ωtI(t) = I_0\cos\omega t. The magnetic field inside the solenoid, B(t)=μ0nI(t)B(t) = \mu_0 n I(t), therefore changes with time. In this problem you will need to make use of wisely chosen Faraday/Amp\`ere loops. (a) Changing BB fields cause EE fields. Assuming that the BB field is given by B0(t)≡μ0nI0cos⁡ωtB_0(t) \equiv \mu_0 n I_0\cos\omega t, find the electric field at radius rr inside the solenoid. (b) Changing EE fields cause BB fields. Find the BB field (at radius rr inside the solenoid) caused by the changing EE field that you just found. More precisely, find the difference between the BB at radius rr and the BB on the axis. Label this difference as ΔB(r,t)\Delta B(r,t). (c) The total BB field does not equal μ0nI0cos⁡ωt\mu_0 n I_0\cos\omega t throughout the solenoid, due to the ΔB(r,t)\Delta B(r,t) difference you just found. What is the ratio ΔB(r,t)/B0(t)\Delta B(r,t)/B_0(t)? Explain why we are justified in making the statement, ``The magnetic field inside the solenoid is essentially equal to the naive μ0nI0cos⁡ωt\mu_0 n I_0\cos\omega t value, provided that the changes in the current occur on a time scale that is long compared with the time it takes light to travel across the width of the solenoid.'' (This time is very short, so for an ``everyday'' value of ω\omega, the field is essentially equal to μ0nI0cos⁡ωt\mu_0 n I_0\cos\omega t.)

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