Question

Magnetic flux linked with a stationary loop of resistance 4 ohm varies with respect to time period 2 seconds, as follows : ∅ = 3t(2-t) . Find the amount of heat generated ( in joules) during that time. ​

Answer 1

the answer is : 18 joules

explanation has been pinned........

Answer 2

The magnetic flux varies with time $\sf{t}$ as,

$\sf{\longrightarrow \Phi=3t(2-t)}$

$\sf{\longrightarrow \Phi=6t-3t^2}$

The emf produced in the loop is,

$\sf{\longrightarrow e=-\dfrac{d\Phi}{dt}}$

$\sf{\longrightarrow e=-\dfrac{d}{dt}\left[6t-3t^2\right]}$

$\sf{\longrightarrow e=6t-6}$

$\sf{\longrightarrow e=6(t-1)}$

Resistance of the loop,

$\sf{\longrightarrow R=4\ \Omega}$

The amount of heat generated in the loop during a very small time $\sf{dt}$ is,

$\sf{\longrightarrow dH=\dfrac{e^2}{R}\ dt}$

Hence total amount of heat generated in the loop during a time period (2 seconds) is,

$\displaystyle\sf{\longrightarrow H=\int\limits_0^2\dfrac{e^2}{R}\ dt}$

$\displaystyle\sf{\longrightarrow H=\dfrac{36}{4}\int\limits_0^2(t-1)^2\ dt}$

$\displaystyle\sf{\longrightarrow H=\dfrac{36}{4}\times\dfrac{1}{3}\left[(t-1)^3\right]_0^2}$

$\displaystyle\sf{\longrightarrow H=3\left(1^3-(-1)^3\right)}$

$\displaystyle\sf{\longrightarrow\underline{\underline{H=6\ J}}}$

Hence 6 J heat energy is generated.