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The voltmeter has infinite resistance. What are the readings on the voltmeter when the switch is open and when it is closed?

I thought that the reading on the voltmeter would be E/2 regardless of whether the switch was open or closed:

When open: The two resistors are identical, so voltage splits evenly;

When closed: The voltmeter has infinite resistance and so it serves as an open switch for that part of the circuit. Even if the voltmeter was connected just around the resistor to the top left of the circuit, while a parallel circuit went around both this resistor and the voltmeter, I would say that the voltage is E/2.

My answer is right for when the switch is opened, but the answer to when the switch is closed is "less than E/2." Why so?

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When the switch is closed the two resistors on the left are in parallel. The current from the battery will split up half passing through one resistor and half passing through the other. The result will be a smaller voltage drop across each resistor, think $V=IR$.

The measured voltage will less than it was with the switch open but not half as much because by adding a second resistor in parallel the total resistance of the circuit is decreased and so the new total current will increase.

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