Question

The amount of heat generated in a resistor of resistance 2 Ω, in a time interval of 2 s, if current in it increases uniformly from zero to 4 A in the 2 s is


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Answer 1

Given :

Resistance of the given resistor = 2 ohm

The time interval in which the current increases from 0 to 4 A = 2 s

To Find :

The heat generated in the resistor in the given time interval = ?

Solution :

The heat generated will be given as :

$dH = i^2Rdt$        - (1)

Here , dH is change in heat , i is current , R is resistance and dt is change in time .

Now we have to write 'i' in terms of t :

∴Change in current in 2 sec = 4 A

Change in current in 1 sec = $\frac{4}{2} = 2$ A

So, change in current in t sec  = 2t

So we can current in terms of t as:

i = 2t

Now putting i = 2t in equation 1 :

$dH = (2t)^2Rdt$

Integrating both sides with proper limits :

$\int\limits^H_0 {} \, dH = \int\limits^2_0 {4t^2R} \, dt$

So, $H = 4R[\frac{t^3}{3}]^2_{0}$

          =$4 \times 2 \times \frac{8}{3}$

          = 21.333 J

Hence , the heat generated in the resistor in 2 sec is 21.33 J.