Question Video: Determining Decrease in Potential for Batteries in Parallel | Nagwa Question Video: Determining Decrease in Potential for Batteries in Parallel | Nagwa

Question Video: Determining Decrease in Potential for Batteries in Parallel Physics • Third Year of Secondary School

The resistor in the circuit shown is powered by two batteries in parallel that are combined in two different configurations. The batteries have e.m.f’s of 3.5 V and 2.5 V respectively. In the first configuration, the positive terminals of the battery are directly connected to each other and the negative terminals are connected to each other. In the second configuration, the positive terminals of each battery are directly connected to the negative terminal of the other battery. Which of the following is a correct statement about how the decrease in potential across the resistor compares in the two configurations? [A] The decrease in potential will be greater in the first configuration. [B] The decrease in potential will be greater in the second configuration. [C] The decrease in potential will be the same in both configurations. [D] The decrease in potential in both configurations will depend on the internal resistances of the batteries.

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Video Transcript

The resistor in the circuit shown is powered by two batteries in parallel that are combined in two different configurations. The batteries have emfs of 3.5 volts and 2.5 volts, respectively. In the first configuration, the positive terminals of the battery are directly connected to each other and the negative terminals are connected to each other. In the second configuration, the positive terminals of each battery are directly connected to the negative terminal of the other battery. Which of the following is a correct statement about how the decrease in potential across the resistor compares in the two configurations? (A) The decrease in potential will be greater in the first configuration. (B) The decrease in potential will be greater in the second configuration. (C) The decrease in potential will be the same in both configurations. (D) The decrease in potential in both configurations will depend on the internal resistances of the batteries.

We have two circuits with different configurations for the parallel batteries, and we want to find the decrease in potential across the resistor. To do this, we will use Kirchhoff’s second law.

Recall that Kirchhoff’s second law states that the sum of the potential difference across each component in a loop is equal to zero. We will begin by considering the first configuration where the positive terminals of the battery are directly connected to each other.

Let’s consider loop one. Applying Kirchhoff’s second law to loop one, we find that 3.5 minus 𝑉 equals zero. Rearranging this gives us 𝑉 equals 3.5 volts. So using this loop, we find that the decrease in potential across the resistor is equal to 3.5 volts.

Now let’s consider loop two. Applying Kirchhoff’s second law to loop two, we find that 2.5 minus 𝑉 equals zero. Rearranging this gives 𝑉 equals 2.5 volts. So using loop two, we find that the decrease in potential across the resistor is equal to 2.5 volts.

We have just calculated two different values for the decrease in potential across the resistor, which doesn’t make any sense. So what’s going on here? Kirchhoff’s second law is a statement of conservation of energy in a circuit. The energy provided by the batteries must be equal to the energy transferred to the resistor. In the calculations we just did, energy can’t have been properly conserved; otherwise we wouldn’t have gotten two different values for the potential difference across this resistor.

In this configuration, there must be an internal resistance in each battery that was not included in our calculations. Some of the energy from each battery can be transferred to these internal resistances, ensuring that energy is conserved in the circuit. If we had considered the potential differences across these internal resistances when we applied Kirchhoff’s second law, we would not have calculated two different values for the decrease in potential across the resistor.

Now let’s consider the second configuration, where the positive terminals of each battery are directly connected to the negative terminal of the other battery. If we consider loop one, then by applying Kirchhoff’s second law to this loop, we find that minus 3.5 minus 𝑉 equals zero. Rearranging this gives 𝑉 equals negative 3.5 volts, which represents the decrease in potential across the resistor using this loop. Note that minus 𝑉 equals minus negative 3.5 volts, which equals a positive number overall. Now let’s consider loop two. Applying Kirchhoff’s second law to loop two, we find that 2.5 minus 𝑉 equals zero. Rearranging this gives 𝑉 equals 2.5 volts, which represents the decrease in potential across the resistor using loop two.

Again, we have just calculated two different values for the decrease in potential across the resistor. So again, we see that there must be an internal resistance in each battery that causes the drop in potential required to conserve energy in the circuit. Without knowing the internal resistances of the batteries, we can’t say for certain how the decrease in potential across the resistor changes between these two configurations. We simply don’t have enough information to complete the calculations.

Therefore, options (A), (B), and (C) are incorrect. The only way to know the exact effect of changing the configuration of the batteries is if the internal resistances of the batteries are known. Therefore, the answer is option (D); the decrease in potential in both configurations will depend on the internal resistances of the batteries.

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