What will happen to the potential difference between the two plates of a parallel plate capacitor in the given situation?

Electric Potential Energy (U) and Electric Potential (V): (Notes from C. Erkal’s lectures PHYS 221)

Consider a parallel plate capacitor that produces a uniform electric field between its large plates.  This is accomplished by connecting each plate to one of the terminals of a power supply (such as a battery).

Figure 1: An electric field is set up by the charged plates separated by a distance l.  The charges on the plates are +Q and –Q.

Figure 2: An electric charge q is moved from point A towards point B with an external force T against the electric force qE.

Figure 3, 4: When it is moved through a distance d, its potential energy at the point B is qEd relative to the point A.

Figure 5: When released from B (T = 0), it will accelerate toward the lower plate.  As it is moving toward the lower plate, its potential energy decreases and its Kinetic energy increases.  When it reaches the lower plate (where we can choose the Potential energy to be zero), its potential energy at A is completely converted to Kinetic Energy at point B:

Note that qEd is the work done by the field as the charge moves under the force qE from B to A.  Here m is the mass of the charge q, and v is its velocity as it reaches point A.  Here we assumed that electric field is uniform!  Work done by E field:

Let’s remember Kinetic Energy-Work theorem (Work Energy principle):

where we introduced the concept of potential energy and conservative force ( a force under which one can define a potential energy so that the work done only depends the differences of the potential energy function evaluated at the end points).

A rule of thumb for deciding whether or not EPE is increasing:

If a charge is moving in the direction that it would normally move, its electric potential energy is decreasing.  If a charge is moved in a direction opposite to that of it would normally move, its electric potential energy is increasing.  This situation is similar to that of constant gravitational field (g = 9,8 m/s2).  When you lift up an object, you are increasing its gravitational potential energy.  Likewise, as you are lowering an object, its gravitational energy is decreasing.

A General Formula for Potential Difference:

The work done by an E field as it act on a charge q to move it from point A to point B is defined as Electric Potential Difference between points A and B:

Clearly, the potential function V can be assigned to each point in the space surrounding a charge distribution (such as parallel plates).  The above formula provides a simple recipe to calculate work done in moving a charge between two points where we know the value of the potential difference.  The above statements and the formula are valid regardless of the path through which the charge is moved.  A particular interest is the potential of a point-like charge Q.  It can be found by simply performing the integration through a simple path (such as a straight line) from a point A whose distance from Q is r to infinity.  Path is chosen along a radial line so that

This process defines the electric potential of a point-like charge.  Note that potential function is a scalar quantity as oppose to electric field being a vector quantity.  Now, we can define the electric potential energy of a system of charges or charge distributions.  Suppose we compute the work done against electric forces in moving a charge q from infinity to a point a distance r from the charge Q.  The work is given by:

Note that if q is negative, its sigh should be used in the equation!  Therefore, a system consisting of a negative and a positive point-like charge has a negative potential energy.

A negative potential energy means that work must be done against the electric field in moving the charges apart!

Now consider a more general case, which deals with the potential in the neighborhood of a number of charges as depicted in the picture below:

Let r1,r2,r3 be the distances of the charges to a field point A, and r12, r13, r23 represent the distance between the charges.  The electric potential at point A is:

Example:

If we bring a charge Q from infinity and place it at point A the work done would be:

The total Electric Potential Energy of this system of charges namely, the work needed to bring them to their current positions can be calculated as follows: first bring q1 (zero work since there is no charge around yet), then in the field of q1 bring q2, then in the fields of q1 and q2 bring q3.  Add all of the work needed to compute the total work.  The result would be:

Finding Electric Field from Electric Potential:

The component of E in any direction is the negative of the rate of change of the potential with distance in that direction:

The symbol Ñ is called Gradient.  Electric field is the gradient of electric potential.  Electric field lines are always perpendicular to the equipotential surfaces.

Equipotentail Surfaces:

These are imaginary surfaces surrounding a charge distribution.  In particular, if the charge distribution is spherical (point charge, or uniformly charged sphere), the surfaces are spherical, concentric with the center of the charge distribution.  Electric field lines are always perpendicular to the equipotential surfaces.   The equation 

A parallel plate capacitor has two conducting plates with the same surface area, which act as electrodes. One plate acts as the positive electrode, while the other one acts as the negative electrode when a potential difference is applied to the capacitor. The two plates are separated by a gap that is filled with a dielectric material. Dielectric materials are electrically insulating and non-conducting, which means that they do not conduct current and can hold the electrostatic charges while emitting minimal energy in the form of heat or leakage currents.

Dielectric materials have the ability of electric polarisation.

Electric polarisation is the tendency of a material’s molecules to obtain an electric dipole moment when the material is placed in an external electric field.

How does a parallel plate capacitor work?

The electrical charges of the material are separated proportionally to the electrical field, creating two poles, a negative and a positive one.

The electric polarisation process is similar to magnetisation, where a magnetic dipole is induced in a magnetic material when placed near a magnet.

Therefore, when dielectric materials are placed in an external electrical field, the dipole moment that is induced per unit volume of the dielectric material is also known as electric polarisation. This is described by the equation below, where k is the dimensionless dielectric constant, E the permittivity of the material, and Eo the permittivity of vacuum, which is around 8.85 × 10-12 farad per metre (F/m).

The two plates of the parallel plate capacitor are connected to a power supply. The plate that is connected to the positive terminal of the battery acquires a positive charge, while the plate that is connected to the negative terminal acquires a negative charge. This happens because the positive pole pushes electrons to the opposite plate. Due to the attraction between the positive and negative charges acquired in the positive and negative plates, the charges are stored within the plates of the capacitor.

Electric field lines are formed between the two plates from the positive to the negative charges, as shown in figure 1. The polarisation of the dielectric material of the plates by the applied electric field increases the capacitor’s surface charge proportionally to the electric field strength in which it is placed.

Figure 1. Parallel plate capacitor configuration. Source: toppr.com.

Parallel plate capacitor: Derivation

The two plates of a parallel plate capacitor are separated by a distance d measured in m, which is filled with atmospheric air. The cross-sectional area of each plate A is measured in m2. The electric field E of each plate is equal to the following, where σ is the surface density.

If the potential difference between the two plates is equal to V, when we substitute the equation found for the electric potential, we get:

Now, substituting the capacitance in the derived voltage, we get:

It can be seen that the capacitance depends on the distance between the plates. The charge stored is proportional to the surface area and inversely proportional to distance. This can also be validated by considering the characteristics of the Coulomb force, where like charges repel and unlike charges attract each other. The force between charges decreases with distance. The bigger the plates, the greater the charge storing capacity as the charges spread out more. Thus, the storable charge is increased when the area is also increased. Similarly, the closer the plates, the greater the attraction force between the opposite charges, so capacitance should be greater when the distance is decreased.

Parallel Plate Capacitor: Leakage currents

A given charge is supplied to each plate. Because there is no ideal dielectric material that can hold the charge perfectly, the increase in the potential leads to leakage currents, which cause the capacitor to discharge in an unwanted way once it is disconnected from the circuit.

The amount of time a capacitor can hold a charge depends on the quality of the dielectric material used in the capacitor.

Parallel plate capacitor: Electric field

In a parallel plate capacitor, when a voltage is applied between two conductive plates, a uniform electric field between the plates is created. However, at the edges of the two parallel plates, instead of being parallel and uniform, the electric field lines are slightly bent upwards due to the geometry of the plates. This is known as the fringing or edge effect (see figure 2).

A capacitor’s electric field strength is directly proportional to the voltage applied while being inversely proportional to the distance between the plates.

Figure 2. Diagram showing the fringing of the electric field at the edges of the two plates.

Usage of parallel plate capacitors

The usage of capacitors range from filtering static out of radio reception to energy storage in heart defibrillators and include the following:

• Energy storage and circuit protection against unusual spikes in voltage or any interruption in the circuit.
• Electronic signal processing.
• Suppression and coupling.
• Motor starters used in pumps and compressors such as refrigerator compressors.

The reason capacitors cannot be used like batteries is that they cannot hold energy for a long time due to the leakage currents.

You can make a parallel plate capacitator at home using two sheets of paper, which are glued together, with an aluminium foil sheet on glued to each side of the paper. Then you need to attach copper wires to the upper right and bottom left corners and connect each wire to the electrodes of a battery.

A parallel plate capacitor has a capacitance of 5 mF. Determine the capacitance after the distance between them is reduced to a third of the initial distance, and with the space between the two plates having a dielectric constant of 7.

Solution:

We derive an expression relating the given capacitance and the new capacitance with the reduced distance.

Determine the area of the capacitor if the potential difference between the plates is 0.5 V, the distance between the plates is 3mm, and a charge of 1.2 ⋅ 10-9 C is stored in the capacitor.

Solution:

We use the equation that relates the potential difference with the area. Then we substitute using the given values in SI units.

Parallel Plate Capacitor - Key takeaways

• A parallel plate capacitor is a device that stores charge.
• Parallel plate capacitors feature two plates made from conductive materials.
• Capacitors store charge by electric polarisation.
• Parallel plate capacitors have a wide range of applications, such as motor starters, signal processors, compressors, etc.

A parallel plate capacitor is a type of capacitor that is constructed by two parallel conducting plates and a dielectric material between them.  It can be used to store electrical energy and signal processing.

We can increase the capacitance of a parallel plate capacitor by increasing the area of the plates or decreasing the distance between the plates.

A parallel plate capacitor stores electrical charges when there is a voltage difference between the plates. Because there is a dielectric material between the plates, the electrical charges will be stored in the dielectric material.

Question

What is a parallel plate capacitor?

Answer

A capacitor is a device used to store electric charge.

Question

List three applications of a parallel plate capacitor.

Answer

• Energy storage.
• Electronic signal processing.
• Pulsed power and weapons.

Question

Which of the following applications is not an application for a parallel plate capacitor?

Answer

Question

What is the main working principle of a parallel plate capacitor?

Answer

Question

What is electric polarisation?

Answer

It is the tendency of a material’s molecules to obtain an electric dipole moment when the material is placed in an external electric field.

Question

What materials are the plates made of?

Answer

Question

What are dielectric materials?

Answer

Materials that have the ability of electric polarisation.

Question

What is the configuration of a parallel plate capacitor?

Answer

A parallel plate capacitor consists of two identical conducting plates connected to the electrodes of a battery. The two plates are separated by a gap that features a dielectric material.

Question

What is a charge leakage?

Answer

When the amount of the supplied charge exceeds a certain limit, the potential increases, which could potentially lead to a leakage in the charge.

Question

Answer

It is the divergence of the electric field lines around the edges of the plates.

Question

How can we construct a parallel plate capacitor?

Answer

Using two aluminium foil layers sandwiched between two paper sheets.

Question

What charge is stored in a 100 μF capacitor when 120 V is applied to it?

Answer

Question

Calculate the voltage applied to a 3 μF capacitor when it holds 5 μC of charge.

Answer

Question

How would you decrease the energy capacity of a capacitor?

Answer

By decreasing the area or increasing the distance between the two plates.