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.
