The total heat produced in resistor R in an R | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) The power dissipated in the resistor is given by $P = I^2 R$.Since the current through the resistor varies with time,we must integrate the power o

Vedclass · Accepted Answer

$\frac{1}{2}LI_0^2$

Vedclass · Answer

(B) The power dissipated in the resistor is given by $P = I^2 R$.Since the current through the resistor varies with time,we must integrate the power over time to find the total heat energy $W$.$W = \int_{0}^{\infty} I^2 R \, dt$The current in an $RL$ circuit during decay is given by $I = I_0 e^{-(R/L)t}$.Substituting this into the integral:$W = \int_{0}^{\infty} (I_0 e^{-(R/L)t})^2 R \, dt = \int_{0}^{\infty} I_0^2 R e^{-(2R/L)t} \, dt$$W = I_0^2 R \left[ \frac{e^{-(2R/L)t}}{-(2R/L)} \right]_{0}^{\infty} = I_0^2 R \left( 0 - \left( -\frac{L}{2R} \right) \right) = \frac{1}{2} L I_0^2$.Alternatively,by the law of conservation of energy,the total heat produced in the resistor must equal the total energy initially stored in the magnetic field of the inductor,which is $U = \frac{1}{2} L I_0^2$.

The total heat produced in resistor $R$ in an $RL$ circuit when the current in the inductor decreases from $I_0$ to $0$ is

Similar Questions

The circuit shown consists of a switch $S$,a battery $B$ of emf $E$,a resistance $R$,and an inductor $L$. What is the current in the circuit at the instant the switch $S$ is closed?

When the switch $S$ is closed at time $t = 0$,what will be the nature of the graphs for the induced $emf$ $e$ across the inductor $L$ and the current $i$ in the circuit?

If $L$ is the inductance and $R$ is the resistance,then the $SI$ unit of $\frac{L}{R}$ is

In an $LR$ circuit,the time constant is the time in which the current grows from zero to the value (where ${I_0}$ is the steady-state current):

$A$ coil of inductance $0.20 \, H$ is connected in series with a switch and a cell of $emf$ $1.6 \, V$. The total resistance of the circuit is $4.0 \, \Omega$. What is the initial rate of growth of the current when the switch is closed? (in $A \, s^{-1}$)

Vedclass Test Series

Exam Paper Generator

Online Exam Module