Question

A battery with internal resistance \(R_{\mathrm{i}}=0.1415 \Omega\) is being charged by a battery charger that delivers a current \(i=5.399 \mathrm{~A}\). The battery charger supplies an emf of \(14.51 \mathrm{~V}\). What is the potential difference across the terminals of the battery?

Short answer

Answer: The potential difference across the terminals of the battery is approximately 13.746 V.

Step-by-step solution

Write down the Ohm's law formula and the formula for the total potential difference

Ohm's law states that the potential difference (\(V\)) across a resistor (\(R\)) is equal to the product of the current (\(I\)) and the resistance (\(R\)). Thus, we can express Ohm's law as: V = I * R In addition to this, the total potential difference can be expressed as the sum of the potential difference across the internal resistance and the potential difference across the terminals of the battery: V_total = V_battery + V_internal

Calculate the potential difference across the internal resistance

Using Ohm's law, we can calculate the potential difference across the internal resistance of the battery. We are given the internal resistance \(R_i = 0.1415\ \Omega\) and the current \(i = 5.399\ \mathrm{A}\). Let's plug these values into the formula: V_internal = i * R_i V_internal = 5.399\ \mathrm{A} * 0.1415\ \Omega V_internal = 0.764\ \mathrm{V} (rounded to three decimal places)

Calculate the potential difference across the battery terminals

We are given the total emf supplied by the charger as \(14.51\ \mathrm{V}\). We can now use the formula for the total potential difference to calculate the potential difference across the battery terminals: V_total = V_battery + V_internal Rearranging the formula to solve for the potential difference across the battery terminals, we get: V_battery = V_total - V_internal Now, let's plug in the values we have: V_battery = 14.51\ \mathrm{V} - 0.764\ \mathrm{V} V_battery = 13.746\ \mathrm{V} The potential difference across the terminals of the battery is approximately \(13.746\ \mathrm{V}\).

Key concepts

Internal Resistance

Internal resistance refers to the inherent resistance to the flow of electric current within a battery or any other electrical source. It is symbolized as \( R_i \) and is measured in ohms (\( \Omega \)). Imagine the battery as a busy highway; internal resistance would be the equivalent of traffic congestion. The higher the internal resistance, the more difficult it is for the current to pass, similar to how heavy traffic slows down the movement of vehicles.

When a battery is in use, its internal resistance causes a small portion of the energy to be lost as heat, slightly reducing the voltage that is available from the battery's terminals. This is why the actual voltage at the terminals (\(V_{battery}\)) is often less than the electromotive force (emf) supplied by the battery. Using our exercise example, when the battery charger with emf of \(14.51\ \mathrm{V}\) is delivering a current of \(5.399\ \mathrm{A}\) through a battery with an internal resistance of \(0.1415\ \Omega\), Ohm's law helps us determine the loss in voltage due to this resistance.

Potential Difference

Potential difference, measured in volts (V), is essentially the voltage across two points in a circuit. It represents the work done to move a unit charge from one point to another. The more formal definition states that the potential difference between two points is the energy required to move a charge from one point to another, divided by the magnitude of the charge.

Applying Ohm's Law

According to Ohm's law, the potential difference across a resistor is directly proportional to the current flowing through it and the resistance of the resistor. This relationship is given by the simple equation \( V = I \times R \). If we know the current flowing and the resistance, we can calculate the potential difference. In our exercise, the potential difference across the battery's internal resistance is found by multiplying the current by the internal resistance. It's important for students to understand that the potential difference across the battery terminals differs from the supplied emf due to the presence of internal resistance.

Electromotive Force (emf)

Electromotive force, abbreviated as emf, and denoted by \( \mathcal{E} \), is the energy provided by a power source to move a unit charge through a circuit. Think of it as the 'pressure' pushing the electrons to move through the circuit, analogous to water pressure pushing water through pipes. Emf is measured in volts and is often considered the 'ideal' voltage of a battery without any internal resistance.

However, the actual voltage available to an external circuit is less than the emf because of the internal resistance within the source. The equation to find the 'actual' or terminal voltage \(V_{battery}\) is \(V_{battery} = \mathcal{E} - V_{internal}\), where \(V_{internal}\) is the voltage lost across the internal resistance. In our exercise, subtracting the voltage drop across the internal resistance from the supplied emf gives us the actual voltage available to do work in the external circuit.