Question

Express the heat produced in calories, when I amperes are passed for t seconds through a resistor having the resistance of R ohms.

Answer 1

Concept used: Electric current

Prerequisite: Knowledge of heat transfer

Step 1: Given data,

1. The current passed through the resistor is I Ampere.
2. The resistance of the resistor is R Ohm.
3. The time taken is t second.

Step 2: Work done by the charge is,

The expression for the charge in terms of current is given by,

$Q=It$

Here, Q is the charge.

The expression of voltage according to Ohm's law is,

$V=IR$

Here, V is the voltage.

The work done by the charge flowing in the resistance is given by,

$W=QV$

Substitute the expression of charge and voltage in the above equation,

$$\begin{gathered} W=\left(It\right)\left(IR\right) \\ W=I^{2}Rt \\ W={\left(I\mathrm{Ampere}\right)}^2\left(R\mathrm{Ohm}\right)\left(t\mathrm{second}\right) \\ W=I^{2}Rt\mathrm{Joule}......\left(1\right) \end{gathered}$$

Step 3: Calculate the heat in calories,

The relation between Calorie and Joule is given by,

$\begin{array}{rcl}1\mathrm{Calorie}& =& 4.184\mathrm{Joule}\\ 1\mathrm{Joule}& =& \frac{1}{4.184}\mathrm{Calorie}\end{array}$

Substitute in equation (1),

$\begin{array}{rcl}W& =& I^{2}Rt\times \frac{1}{4.184}\mathrm{Calorie}\\ W& =& \frac{I^{2}Rt}{4.184}\mathrm{Calorie}\end{array}$

The amount of work done by the current will be equal to the heat produced by the resistor as the energy is conserved for the resistor; therefore, the amount of electrical energy supplied will be equal to the thermal energy.

The heat produced by the resistor will be,

$\begin{array}{rcl}H& =& W\\ H& =& \frac{I^{2}Rt}{4.184}\mathrm{Calorie}\end{array}$

Thus, the heat produced by the resistor in calories is $\frac{I^{2}Rt}{4.184}$.