Question

You want to make an ohmmeter to measure the resistance of unknown resistors. You have a battery with voltage \(V_{\mathrm{emf}}=9.00 \mathrm{~V},\) a variable resistor, \(R,\) and an ammeter that measures current on a linear scale from 0 to \(10.0 \mathrm{~mA}\) a) What resistance should the variable resistor have so that the ammeter gives its full-scale (maximum) reading when the ohmmeter is shorted? b) Using the resistance from part (a), what is the unknown resistance if the ammeter reads \(\frac{1}{4}\) of its full scale?

Short answer

Answer: The variable resistor should be set to 900Ω to give the ammeter its maximum reading when the ohmmeter is shorted. When the ammeter reads 1/4 of its full scale, the unknown resistance is 2700Ω.

Step-by-step solution

Part (a) - Calculate the resistance of the variable resistor

When the ohmmeter is shorted, the ammeter and the variable resistor are connected in series to the battery. To find the resistance value (R) that makes the ammeter show its maximum reading, we can use Ohm's Law: V = IR Where V is the voltage across the battery (9V), I is the maximum current that the ammeter can read (10mA or 0.01A), and R is the resistance of the variable resistor. Rearranging to solve for R: R = V/I Plugging in the given values: R = 9V / 0.01A

Calculate the resistance of the variable resistor (R)

R = 900Ω The variable resistor should be set to 900 ohms to give the ammeter its maximum reading when the ohmmeter is shorted.

Part (b) - Determine the unknown resistance

When the ammeter reads 1/4 of its full scale, it is measuring a current of 2.5mA (or 0.0025A). We can use the voltage divider formula to find the unknown resistance (R_unknown) value: I = V/(R + R_unknown) For this case, we know that I = 0.0025A, V = 9V (battery voltage), and R = 900Ω. So: 0.0025A = 9V/(900Ω + R_unknown) Rearranging to solve for R_unknown: R_unknown = (9V/0.0025A) - 900Ω

Calculate the unknown resistance (R_unknown)

R_unknown = (3600Ω) - 900Ω R_unknown = 2700Ω So, when the ammeter reads 1/4 of its full scale, the unknown resistance is 2700 ohms.