Question

A square conducting loop of side L contains a resistor, R. It is placed in a magnetic field, B, directed into the page and which changes as a function of time t according to the relationship B = at + b, where a and b are positive constants. No magnetic field exists outside the loop.

Derive an expression for the induced current in the loop in terms of a, b, L, and/or R (note: not all terms may be in your final expression).
Derive a similar expression in terms of a, b, L, and/or R for the power dissipated in the resistor.
Does the induced current flow clockwise or anticlockwise? Explain.
The square loop is replaced by a circular copper loop of diameter 0.20 m and consisting of 50 turns. If a = 10 and b = 2, determine the emf induced in the circular loop.

Answer 1

$Solution.\\1)\;the\; area\; of\; a\; square\;S=L^2\\i=-\frac{S}{R}\times\frac{dB}{dt};\\i=-\frac{L^2}{R}\times a ;\\2)\;the \;current\; is \;constant, so\\P=I^2R\;orP=a^2R\\3)\;The \;current\; flows\; counter\;clockwise.\\the\; induction\; current\; creates \\a\; field\; directed \;against\; the\; external\; field.\\4)E_i=-S\times\frac{dB}{dt}\times n;\\E_i=-\frac{\pi\times d^2}{4}\times\frac{dB}{dt}\times n;\\E_i=-0.0314\times 10\times 50=15.7V$