Question

What amount of heat will be generated in a coil of resistance R due to a charge q passing through it, if the current in the coil decreases down to zero uniformly during a time interval $\Delta$ t?

• A. $\frac{4}{3}\frac{q^{2}R}{\Delta t}$
• B. $\frac{2}{3}\frac{q^{2}R}{\Delta t}$
• C. $\frac{3}{4}\frac{q^{2}R}{\Delta t}$
• D. $\frac{3}{2}\frac{q^{2}R}{\Delta t}$

Answer 1

The correct option is A $\frac{4}{3}\frac{q^{2}R}{\Delta t}$

Suppose initial current is $i_0$, then
$i\left(t\right)=i_0(1-\frac{t}{\Delta t})q=i_0{\int }_0^{\Delta t}(1-\frac{t}{\Delta t})dt$
So, $i_0=\frac{2q}{\Delta t}$
$H={\int }_0^{\Delta t}{\left\{\frac{2q}{\Delta t}(1-\frac{t}{\Delta t})\right\}}^{2}Rdt$