### Video Transcript

The diagram shows the scale of an ohmmeter that is being used to measure an unknown resistance. The resistance of the ohmmeter is 25 kilohms. The angle of full-scale deflection of the ohmmeter is Φ equals 60 degrees. The angle of deflection of the ohmmeter arm, 𝜃, equals 30 degrees. What is the unknown resistance? Answer to the nearest kilohm.

In our diagram, we see the scale of an ohmmeter, a device for measuring resistance. Interestingly, this scale is directly measuring current rather than resistance. The electrical circuit, which altogether functions as an ohmmeter, may look like this. This circuit consists of a constant voltage supply of fixed resistor and an ammeter, a device for measuring current. The total resistance of all the resistive components in this circuit is equal to 25 kilohms. When charge flows through this circuit under these conditions, we know that the needle or the arm on the ammeter is fully deflected.

With no unknown resistor to measure then, under normal operation, this ammeter measures a full-scale current, the maximum current its scale will allow. However, if we then take some unknown resistor and connect it in series in our circuit, then the measurement arm on our scale deflects back through an angle of 30 degrees. That is, when we didn’t have our unknown resistor as part of our circuit, our measurement arm experienced full-scale deflection. But then, when we did insert the unknown resistor, our measurement arm now deflects at half the full scale.

Values of current read off of an ammeter scale relate to resistance through the equation known as Ohm’s law. This law says that the voltage in a circuit is equal to the current in the circuit multiplied by the circuit’s resistance. In our circuit, which is functioning as an ohmmeter, our voltage supply is constant. And that means that if current in the circuit 𝐼 for example decreases, then that must reflect an overall increase in the resistance of the circuit. 𝐼 times 𝑅 must always equal the same value of 𝑉. When the ohmmeter arm was deflected to its full-scale reading, in other words before we put our unknown resistor into the circuit, we can say that the current in the circuit then is 𝐼 one.

Let’s also label the resistance of the ohmmeter, we’ll call it 𝑅 sub o, and that’s 25 kilohms. Using Ohm’s law, we can say that 𝑉 which is the voltage supplied by our cell is equal to 𝐼 one times 𝑅 sub o. Then, when we do insert our unknown resistor and the ohmmeter arm deflects to this middle point, let’s say that then the current in our circuit is 𝐼 two. If the resistance of our unknown resistor is 𝑅 sub u, then we can write another application of Ohm’s law for the case when this unknown resistor is part of our circuit. The same potential difference 𝑉 is equal to the new current in the circuit 𝐼 two multiplied by the sum of the original resistance of the ohmmeter plus the resistance of the unknown resistor.

Note that we add these two resistances together because that’s how resistors in series combine. Note that the potential difference 𝑉 in both of these equations is the same. Therefore, we can set the right-hand side of each of these two equations as being equal to one another. As we’ve seen, a full-scale deflection of our ohmmeter arm indicates the current 𝐼 one. On the other hand, a half-scale deflection indicates the current 𝐼 two. This tells us that the current 𝐼 one is twice as great as the current 𝐼 two. We can therefore replace 𝐼 sub one in this equation at the bottom of our screen with two times 𝐼 sub two.

And notice that now the factor of 𝐼 two appears on both sides of the equation. We can therefore divide both sides of the equation by the current 𝐼 two. And the result is that 𝐼 two cancels out completely. We find then that two times 𝑅 sub o, the resistance of the ohmmeter, equals the resistance of the ohmmeter plus the resistance of the unknown resistor. If we subtract 𝑅 sub o from both sides, then on the left, two times 𝑅 sub o minus 𝑅 sub o equals simply 𝑅 sub o, and on the right, 𝑅 sub o minus 𝑅 sub o is equal to zero. We find then that the resistance of the ohmmeter is equal to the resistance of the unknown resistor. That resistance we know is 25 kilohms. This is the resistance of the unknown resistor to the nearest kilohm.