Question

A steady potential difference of $\;100V$ produces heat at a constant rate in a resistor. The alternating voltage which will produce half the heating effect in the same resistor will be:-
(A) $\;100V$
(B) $\;50V$
(C) $\;70.7V$
(D) $\;141.4V$

Answer 1

Hint: Electric potential could be a location-dependent amount that states the number of potential energy per unit of charge at such a location. Once a coulomb of charge (or any given quantity of charge) possesses a comparatively great quantity of potential energy at a given location, then that location is alleged to be a location of high electric potential.
Formula used:
Heat produced,
$H = \dfrac{{{V^2}t}}{R}$
where, $H$ is the heat produced, $V$ is the voltage, $t$ is the time and $R$ is the resistance.

Complete step-by-step solution:
In this question, the heat is altered and that causes the heat produced by is half of its initial.
So they are questioning the factor for which the applied potential difference changes.
Suppose the original potential difference applied to be $V$ and the original heat produced will be $H$.
Then,
$H' = \dfrac{{{{V'}^2}t}}{R}$
According to the question statement,
$ \Rightarrow H' = \dfrac{H}{2}$
Now we will put the values of both the heat,
 We get
$ \Rightarrow \dfrac{{{{V'}^2}t}}{R} = \dfrac{1}{2} \times \dfrac{{{V^2}t}}{R}$
On further solving this equation, we get
$ \Rightarrow {V'^2} = \dfrac{{{V^2}}}{2}$
Which implies,
$ \Rightarrow V' = \dfrac{V}{{\sqrt 2 }}$
Substitute the value of the original potential difference $V = 100V$ in the above equation we get,
$ \Rightarrow V' = \dfrac{{100}}{{\sqrt 2 }}$
It is the RMS value of voltage. Therefore, we have to convert it into its peak value
$ \Rightarrow {V_{rms}} \times \sqrt 2 = \dfrac{{100}}{{\sqrt 2 }} \times \sqrt 2 $
$ \Rightarrow 100V$

Hence the correct answer is option (A) $\;100V$ .

Note: The key concept is why heat is generated when a charge is displaced in potential from one point to another. The potential difference acting between the points in which the charge is being displaced does some work over the charge and this work done constitutes heat formation of heat. The basic definition of potential difference itself suggests that it is work done per unit charge.