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Chapter 26: Problem 58

An ammeter with an internal resistance of \(53 \Omega\) measures a current of \(5.25 \mathrm{~mA}\) in a circuit containing a battery and a total resistance of \(1130 \Omega\). The insertion of the ammeter alters the resistance of the circuit, and thus the measurement does not give the actual value of the current in the circuit without the ammeter. Determine the actual value of the current.

Short Answer

Expert verified
Answer: The actual value of the current in the circuit without the ammeter is approximately 5.44 mA.

Step by step solution

01

Calculate the original resistance of the circuit without the ammeter

Without the ammeter, the total resistance of the circuit is \(R = 1130 \Omega\). We want to find the actual current without the ammeter.
02

Calculate the equivalent resistance of the circuit with the ammeter

The internal resistance of the ammeter is \(R_A = 53 \Omega\). When the ammeter is inserted into the circuit, the equivalent resistance of the circuit becomes the sum of the original resistance and the ammeter's resistance: \(R_{eq} = R + R_A\).
03

Calculate the actual current in the circuit

The ammeter measures a current of \(5.25 \mathrm{mA}\) in the circuit with an altered resistance of \(R_{eq}\). Let \(I_{measured}\) be the measured current and \(I_{actual}\) be the actual current we want to find. We can apply Ohm's Law (V=IR) to both the circuit with and without the ammeter: \(I_{actual} R = I_{measured} R_{eq}\) Now, we need to solve for \(I_{actual}\). Since we know the value of the resistances and the measured current, we can plug in values: \(I_{actual} = \frac{I_{measured} R_{eq}}{R}\)
04

Plug in values and solve for the actual current

We can now plug in the values for the measured current and resistances into the formula: \(I_{actual} = \frac{(5.25 \mathrm{~mA})(1130 \Omega + 53 \Omega)}{1130 \Omega}\) \(I_{actual} = \frac{5.25 \mathrm{~mA} \cdot 1183 \Omega}{1130 \Omega}\) \(I_{actual} \approx 5.44 \mathrm{~mA}\) Therefore, the actual value of the current in the circuit without the ammeter is approximately \(5.44 \mathrm{~mA}\).

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