25.5
Energy Stored in a Capacitor
When an archer pulls the string in a bow, he saves the work done in the form of elastic potential energy. When he releases the string, the potential e…
Consider a parallel plate capacitor connected to a battery. Work is done to move the electrons such that a potential difference is developed across the plates.
Suppose, at time t, the plates have acquired charge q, the potential difference across the plates is expressed as the ratio of the acquired charge to the capacitance of the capacitor.
Now, to increase the charge on the plates by a small amount, additional work done is expressed as the product of the potential difference between the plates and the additional charge acquired.
Integrating the expression for the additional work done within the limits of zero to Q, the total work done to acquire a final charge Q can be obtained.
Now, the potential energy gained by the capacitor equals the total work done to acquire charge Q, which can be expressed in terms of potential difference.
Substituting for capacitance and potential difference in terms of electric field, the potential energy per unit volume of the capacitor gives the energy density between the charged capacitor plates.
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Q1: How is work related to the energy stored in a capacitor?
The energy stored in a capacitor equals the total work done to charge it. When a capacitor is connected to a battery, work is performed to move electrons and create a potential difference across the plates. This work is converted into electric potential energy and stored in the capacitor's electric field. The stored energy can be calculated by integrating the work done as charge gradually accumulates on the plates.
Q2: What is the relationship between charge, potential difference, and capacitance during charging?
At any instant during charging, the charge q and potential difference V across a capacitor's plates are related by the equation V = q/C, where C is the capacitance. As the capacitor charges, the charge gradually builds on the plates while the potential difference increases proportionally. This relationship holds throughout the charging process until the capacitor reaches its final charge Q.
Q3: How do you calculate the total potential energy stored in a charged capacitor?
The potential energy stored in a capacitor can be calculated using UC = Q²/2C or UC = ½CV², where Q is the final charge, C is capacitance, and V is the potential difference. These formulas are derived by integrating the work done (dW = V dq) as charge accumulates from zero to Q. Both expressions are equivalent and provide the total electric potential energy stored in the capacitor.
Q4: What is energy density in a charged capacitor?
Energy density is the potential energy per unit volume stored between the capacitor plates. For a parallel plate capacitor with area A and plate separation d, the energy density is expressed as u = ½ε₀E², where E is the electric field and ε₀ is the permittivity of free space. This quantity describes how concentrated the electric potential energy is within the dielectric space between the plates.
Q5: How does the electric field relate to energy density in a capacitor?
The energy density stored in a capacitor is directly proportional to the square of the electric field: u = ½ε₀E². Since the electric field E = σ/ε₀ (where σ is surface charge density), stronger electric fields result in higher energy densities. This relationship shows that energy storage in a capacitor depends fundamentally on the intensity of the electric field between the plates.
Q6: Why is a capacitor analogous to a drawn bow?
Both a capacitor and a drawn bow store energy through work done against a restoring force. When an archer pulls a bowstring, work is saved as elastic potential energy; releasing it converts that energy to kinetic energy of the arrow. Similarly, a capacitor stores work done by a battery as electric potential energy; this energy can be released as electrical current when the capacitor discharges through a circuit.
Q7: How does capacitance affect the amount of energy stored at a given potential difference?
Energy storage is directly proportional to capacitance when potential difference is constant, as shown by UC = ½CV². A capacitor with larger capacitance stores more energy at the same voltage. Since capacitance depends on plate area, plate separation, and the dielectric material between plates, these physical parameters directly influence how much energy the capacitor can store.