# Why is EMF = I(R+r)?

A voltmeter is connected to a dry cell battery. The voltmeter reads 9 V. A 24Ω resistor is connected across the terminals of the battery and the voltmeter reading becomes 7.2 V. Calculate the internal resistance of the cell.

Solution:

The circuit:

We assume the voltmeter draws no current, so in the first experiment we are measuring the EMF of the cell i.e. E = 9.0V.    When we connect a 24Ω resistor we now have a circuit with the 9V cell, the internal resistance r and the external 24Ω resistor.

The voltage measured across the cell is now 7.2V, so that means we have 7V across the 24Ω resistor and therefore the current is I = V/R = 7.2/24 = 0.3A.

Now, for internal resistance:

Equation 1: 9 - 7.2 = 0.3 * r

Equation 2 I am not sure how to write:

$$7.2 = 0.3(24+r) or 9 = 0.3(24+r)$$

I prefer the first; let’s consider the above diagram with arrows as the path for current. Then, the current travels through both the cell and the 24ohm resistor. The internal resistance is present only when there is current flow i.e. when V = 7.2.

Why is the second equation right?

• Why do you need 2 equations, if you can use one already? Nov 18 at 9:36
• @Justme I want to understand every equation. Equation 1 is directly using formula from Textbook & equation 2 , I thought of on my own. Nov 18 at 9:37
• But in equation 2 , i see two types. Therefore , I want to correct where am I wrong. Nov 18 at 9:38
• "the internal resistance is always present" It only has an effect if current flows through it causing a voltage drop. Your ideal voltmeter takes no current and causes no voltage drop in the internal resistance. Hence it reads the EMF. Nov 18 at 11:45
• To help anyone who is interested, the direct link to the chatroom mentioned above is here: chat.stackexchange.com/rooms/131582/chat-between-smt-graham Nov 19 at 16:07

It is difficult to help you when you write your equations with no symbols or units. Your second equation seems to indicate that $$V_R = I_{\text{total}} \cdot (R+r)$$ which is clearly wrong. Your third equation indicates $$V_{\text{total}} = I_{\text{total}} \cdot (R+r)$$ which is correct. Hopefully you can see why. From that, you can deduce what $$\r \$$ should be.