Question

If 4000 J of work is done to transfer 80 C of charge through a circuit having resistance 100 Ω then calculate the time taken for the charges to flow.​

Answer 1

Understanding Concept :-

The heat produced or the work done, "W" by the current I for a given time period "t" is given by the formula.

$\quad\quad\green{ \underline { \boxed{ \sf{W=I^2Rt}}}}$

So, $\boxed{\sf Time\: period,t = \frac{W}{I^2R}}$

Given :-

$\red{\leadsto}\:$$\textsf{Work,W}$$\sf =4000J$

$\green{\leadsto}\:$$\textsf{Resistance,R}$$\sf = 100\:Ω$

$\purple{\leadsto}\:$$\textsf{Charge transfered,q}$$\sf = 80 \;C$

Solution :-

First , finding Voltage,V

$\quad\quad\green{ \underline { \boxed{ \sf{V=\frac{W}{q}}}}}$

$\begin{gathered}\\\\\implies\quad \sf V=\frac{4000}{80} \\\end{gathered}$

$\begin{gathered}\\\\\implies\quad \sf V= 50 \:volts \\\end{gathered}$

Now,

$\quad\quad\red{ \underline { \boxed{ \sf{Current, I =\frac{V}{R}}}}}$

$\begin{gathered}\\\\\implies\quad \sf I = \frac{50}{100}\\\end{gathered}$

$\begin{gathered}\\\\\implies\quad \sf I = \frac{1}{2}\:ampere\\\end{gathered}$

Finally, using the derived formula

$\quad\quad\boxed{\sf Time \:period,t = \frac{W}{I^2R}}$

$\begin{gathered}\\\\\implies\quad \sf t = \frac{4000}{(\frac{1}{2})^2\times100} \\\end{gathered}$

$\begin{gathered}\\\\\implies\quad \sf t = \frac{4000}{\frac{1}{\cancel{4}}\times\cancel{100}} \\\end{gathered}$

$\begin{gathered}\\\\\implies\quad \sf t = \frac{\cancel{4000}}{\cancel{25}} \\\end{gathered}$

$\begin{gathered}\\\\\implies\quad \sf t = 160 s \\\end{gathered}$

Thus, it will take 160s for charges to flow through the circuit.