Question

4 cell each of emf \(2 \mathrm{v}\) and internal resistance of \(1 \Omega\) are connected in parallel to a load resistor of \(2 \Omega\) Then the current through the load resistor is.... (A) \(2 \mathrm{~A}\) (B) \(1.5 \mathrm{~A}\) (C) \(1 \mathrm{~A}\) (D) \(0.888 \mathrm{~A}\)

Short answer

The current through the load resistor is \(0.888 \mathrm{~A}\) (Option D).

Step-by-step solution

Find the equivalent emf and internal resistance of the parallel combination of cells

When cells are connected in parallel, the emf remains the same, but the internal resistance decreases. For 'n' cells connected in parallel, the equivalent internal resistance (R_eq) is given by: \[R_{eq} = \frac{R}{n}\] In our case, there are 4 cells with internal resistance of 1 ohm each, so, \[R_{eq} = \frac{1}{4}\] Thus, the equivalent emf is 2V and the equivalent internal resistance is \(0.25\Omega\).

Use Ohm's law to find the current through the load resistor

Now that we have the equivalent emf and internal resistance, we can treat the parallel combination of cells as a single cell with these values. The total resistance of the circuit will be the sum of the load resistor and the equivalent internal resistance. According to Ohm's law, the current (I) passing through the load resistor would be: \[I = \frac{E_{eq}}{R_{eq} + R_L}\] where \(E_{eq}\) is the equivalent emf (2V), \(R_{eq}\) is the equivalent internal resistance (\(0.25\Omega\)), and \(R_L\) is the load resistor (2 ohms): \[I = \frac{2}{0.25 + 2}\] Calculating this, we get: \[I = \frac{2}{2.25}\] \[I = 0.888 \mathrm{~A}\] Thus, the current through the load resistor is \(0.888 \mathrm{~A}\), which corresponds to option (D).

Key concepts

Ohm's Law

Ohm's Law is a fundamental principle in electric circuits that relates the voltage, current, and resistance within a circuit. According to this law, the current flowing through a conductor between two points is directly proportional to the voltage across the two points and inversely proportional to the resistance of the conductor. The mathematical representation of Ohm's Law is given by the equation:\[ I = \frac{V}{R} \]Where:

• \( I \) is the current in amperes (A),
• \( V \) is the voltage in volts (V), and
• \( R \) is the resistance in ohms (\( \Omega \)).

In the exercise, Ohm's Law is used to determine the current through the load resistor after calculating the equivalent emf and internal resistance of the circuit. By substituting these values into the formula, we can find the current that flows through the circuit.

emf (Electromotive Force)

The emf, or electromotive force, is a measure of the electrical energy supplied by a source such as a battery or a generator. It is defined as the potential difference in volts ( V) provided by the source when no current is flowing in the circuit. This is the maximum potential that the source can provide. The key point to remember is that emf is not truly a "force" but rather a potential difference that has the ability to cause current to flow in a circuit. In the given problem, each of the cells has an emf of 2V. When connected in parallel, the overall emf of the circuit remains the same at 2V. This is because when cells are in parallel, their voltages do not add up; rather, the voltage across each cell remains equal.

Internal Resistance

Internal resistance refers to the resistance within a battery or power source that impedes the flow of current. All practical sources have some internal resistance, although it's ideally something we want to minimize to get maximum current flow to the external circuit. The internal resistance is often denoted as \( R_{int} \).When multiple cells are connected in parallel, the effective internal resistance decreases because the current has multiple pathways to flow. The formula to calculate the equivalent internal resistance for 'n' identical cells connected in parallel is given as:\[ R_{eq} = \frac{R}{n} \]In this exercise, each cell has an internal resistance of 1\( \Omega \). With four cells in parallel, the equivalent internal resistance is \(0.25 \Omega\). This reduced resistance allows more current to flow through the load resistor.

Parallel Circuits

Parallel circuits are a configuration in which components are connected along multiple paths so the same voltage is applied across each component. In a parallel circuit, the total resistance decreases as more components are added. This is because there are additional paths for the current to flow.When dealing with parallel circuits, some important characteristics include:

• The total voltage across each parallel path is the same.
• The total current in the circuit is the sum of the currents through each parallel branch.
• The total or equivalent resistance \( R_{eq} \) is less than the smallest individual resistance in the parallel network.

In the exercise, four cells are connected in parallel, which means that their individual emfs stay the same (2V) across the load. The current path is divided into four, making it easier for the current to flow and reducing the overall resistance effectively.