Question

A battery is used to charge a parallel plate capacitor till the potential difference between the plates becomes equal to the electromotive force of the battery. The ratio of the energy stored in the capacitor and work done by the battery will be (A) \((1 / 2)\) (B) \((2 / 1)\) (C) 1 (D) \((1 / 4)\)

Short answer

The ratio of the energy stored in the capacitor and the work done by the battery is \(\frac{1}{2}\).

Step-by-step solution

Write the formula for energy stored in a capacitor

The formula for the energy stored in a capacitor is given by: \[U_{cap} = \frac{1}{2}CV^2\] where \(U_{cap}\) is the energy stored in the capacitor, \(C\) is the capacitance, and \(V\) is the potential difference across the capacitor.

Write the formula for work done by the battery

The formula for the work done by the battery to charge the capacitor is given by: \[W_{battery} = QV\] where \(W_{battery}\) is the work done by the battery, \(Q\) is the charge on the capacitor, and \(V\) is the potential difference (EMF of the battery in this case) across the capacitor.

Express charge in terms of capacitance and potential difference

Recall that the charge on a capacitor can be expressed as: \[Q = CV\] Substitute this expression for charge into the work done by the battery formula: \[W_{battery} = (CV)V\] \[W_{battery} = CV^2\]

Calculate the ratio of energy stored and work done

Now we can find the ratio of the energy stored in the capacitor to the work done by the battery: \[\frac{U_{cap}}{W_{battery}} = \frac{\frac{1}{2}CV^2}{CV^2}\]

Simplify the ratio

Simplify the expression for the ratio: \[\frac{U_{cap}}{W_{battery}} = \frac{\frac{1}{2}CV^2}{CV^2} = \frac{1}{2}\] According to our calculations, the ratio of the energy stored in the capacitor and the work done by the battery is \(\frac{1}{2}\). Therefore, the correct answer is: (A) \(\frac{1}{2}\)

Key concepts

Capacitor Charging

When a battery is connected to a capacitor, it delivers energy, causing the capacitor to "charge up." This means that the capacitor accumulates an electric charge on its plates until the potential difference across the plates matches the electromotive force (EMF) of the battery. This process is essential for storing energy in the capacitor. The charge is distributed equally until full capacity, dependent on the capacitance and the potential difference.
As the capacitor charges, the electric field between its plates grows stronger. This increasing field results in the accumulation of energy within the capacitor. It's important to note that the charging process is gradual, and the speed at which a capacitor charges can depend on various factors, such as resistance in the circuit and the initial charge of the capacitor.

Work Done by Battery

The work done by a battery is essentially the total energy transferred from the battery to the capacitor. In charging a capacitor, work is required to move charge from one plate to another, overcoming the electrostatic forces in between.
To calculate this work, the formula used is \(W_{battery} = QV\), where \(Q\) is the charge and \(V\) is the potential difference. As the kinetic energy of the charge is negligible, virtually all of this work is then stored as potential energy in the capacitor.
This helps explain why the work done by the battery and the energy stored in the capacitor are closely related and why calculating their ratio, like in the exercise, showcases their direct association.

Electric Potential Difference

Electric potential difference, commonly referred to as voltage, is crucial in understanding capacitors. It's the driving force that moves charge carriers between two points. When a battery connects to a capacitor, it establishes an electric potential difference that prompts electrons to move onto the capacitor plates, resulting in a buildup of charge and stored energy.
The potential difference is key because it determines how much energy can be stored in the capacitor. A greater potential difference generally allows for more charge accumulation, thereby storing more energy within the capacitor systems. This potential is what guides circuits and makes the capacitor beneficial for temporary energy storage.

Capacitance and Charge Relationship

Capacitance is a measure of the ability of a capacitor to store charge per unit potential difference, expressed as \(C = \frac{Q}{V}\). This relationship indicates that capacitance directly connects charge \(Q\) and potential \(V\).
The charge stored, therefore, depends on these two factors: capacitance and potential difference. This relationship means that a capacitor with high capacitance can store more charge even at lower voltages, making it a powerful component in electronic devices for maintaining stable voltage levels.
Understanding how capacitance, charge, and voltage interplay helps in optimizing and using capacitors more effectively in various technological setups and in solving problems like calculating energy storage and discharge times.