An ideal coil of self-inductance and zero resistance carries the current I(t) = I_0 t

PHYSICS · Multi-part · Fermi University Physics · Electricity, Magnetism & Electromagnetism · Electromagnetic induction and alternating-current circuits · Electromagnetic induction and alternating-current circuits

Electricity, Magnetism and Electromagnetism, Ch AC circuits: phasors, complex impedance, resonance and power

Problem

An ideal coil of self-inductance

L

and zero resistance carries the current

I(t) = I_0\cos\omega t ,

and the voltage across it in the direction of the current is

V = L\,dI/dt

. ((a)) Using the complex representation

I = \mathrm{Re}[\tilde I\mathrm{e}^{i\omega t}]

, in which

d/dt

becomes multiplication by

i\omega

, derive

\tilde V = i\omega L\tilde I

. Hence show that the amplitude ratio is

X_L = \omega L

and that the current lags the voltage by a quarter cycle, and confirm the phase by differentiating the real cosine directly. ((b)) Evaluate

X_L

for

L = 25\,\mathrm{mH}

f = 400\,\mathrm{Hz}

. ((c)) With

I_0 = 0.30\,\mathrm{A}

at that frequency, find the voltage amplitude. ((d)) Write

V(t)

in the form

V_0\cos(\omega t + \phi)

and state

\phi

in degrees.

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