Question

The voltage across the terminals of a $9.0V$battery is $8.5V$ when the battery is connected to a $20\Omega$ load. What is the battery’s internal resistance?

Short answer

The battery's internal resistance is$r=1.18\Omega$.

Step-by-step solution

Given Information  

We have to find the internal resistance of the battery.

Simplify 

The resistor $20\Omega$is connected to the battery, a voltage drop occurs which is $8.5V$. This is the potential difference across the resistor role="math" localid="1648884553842" $△V_R=8.5V$. So, determine the current through the circuit by using Ohm's law by

$I=\frac{△V_R}{R}=\frac{8.5V}{20\Omega }=0.425A$

A battery that is connected to a circuit, the current that flows through the circuit is given by equation in the form

role="math" localid="1648888679002" $I=\frac{\epsilon }{R+r}\left(1\right)$

Where $\epsilon$is the emf of the battery which is $9V$. Solve the equation $\left(1\right)$for $r$to be

$$\begin{gathered} I=\frac{\epsilon }{R+r} \\ IR+Ir=\epsilon \left(2\right) \\ r=\frac{\epsilon -IR}{I} \end{gathered}$$

Putting the values for $\epsilon ,I$and$R$into equation$\left(2\right)$to get $r$by

role="math" localid="1648888910241" $$\begin{gathered} r=\frac{\epsilon -IR}{I} \\ =\frac{9V-(0.425A)\left(20\Omega \right)}{0.425A} \\ =1.18\Omega \end{gathered}$$