Electrostatic Potential and Capacitance: Introduction & Derivations

Electric potential and capacitance are related to each other as capacitance measures the capacity to store charge, whereas electric potential measures the ability to do work on a charge.

• Electric Potential measures the ability to perform work on a charge and Capacitance measures the ability to store charge.
• It determines the flow of charge from one body to another body.
• The unit of measurement for capacitance is Farads (f) or Coulomb per Voltage (C/V).
• It is the amount of charge that exists per voltage applied.
• A surface at every point of which electric potential due to charge distribution is the same is called an equipotential surface.

Key Terms: Electrostatics, Electrostatic Potential, Capacitors, Electric Field, Electric Charge, Capacitance, Van De Graff Generator, Voltage, Coulomb, Electrostatic Phenomenon, Combination of Capacitor, Effect of Dielectric on Capacitance

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What is Electrostatics?

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Electrostatics is a branch of physics dealing with the features of electric charges that are either slow-moving or stationary.

A few examples of Electrostatic Phenomenon include:

• Laser machine and photocopier
• The spontaneous explosion of grain silos
• The attraction of paper to a charger scale

Electrostatics

In schools, we all have rubbed the scale on our heads to attract paper from that scale.

• It is one such example of an Electrostatic Phenomenon.
• The electrostatic potential is the force that is external but conservative.
• The work done by an external force that brings a charge from point r to point p.
• Electrostatic is the difference in the potential energy of charges between initial and final points.

Also Read:

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Electrostatic Charge

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We all know about the charge that flows through conductors that we simply call "electricity". The charge can also be present on insulators and since these materials don’t allow the charge to flow, this is being referred to as an Electrostatic Charge.

Two types of charges include:

• Positive Charge
• Negative Charge

The charge depends upon the addition or removal of electrons from an atom. In case there are more protons as compared to electrons, then in this case atoms will be positively charged while in case there are more electrons than protons, the atom will be negatively charged.

Electrostatic Charge

Electrostatic Potential

The potential of a point is defined as the work done per unit charge that results in bringing a charge from infinity to a certain point.

Some things to know about electric potential:

• Electrostatic potentials are mentioned by V and it is a scalar quantity.
• The electrostatic potential is measured in volts.

Electrostatic Potential Difference

The electrostatic potential among 2 points in an electric field is defined as the amount of work done by them in moving a unit positive test charge from one point to another against electrostatic force without any acceleration.

It is being defined as the difference of electrostatic potentials of the two points in the electric field.

Equipotential Surfaces

It is a type of surface over which potential has a constant value. When seen as a point charge, concentric spheres centered at a fixed location of the charge are equipotential surfaces.

Potential Energy

The potential energy stored in a system is considered as the work done by the external influence while arranging the charges in their locations.

Polarisation

The plates of a capacitor enclose a medium that can be filled with insulating substances, and then a net dipole moment is induced by the electric field in the dielectric. Polarisation increases in the field in the opposite direction.

Capacitor

A capacitor is a system of two conductors separated by an insulating material.

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Electric Potential

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Electric Potential is the amount of work needed to move a unit charge from a reference point to a specific point against an electric field.

• The reference point is Earth, although any point that falls beyond the influence of the electric field charge can be used. 
• Electric potential difference is the external work needed to bring a charge from one location to another location in an electric field.

Electric Potential Energy

Unit for Electric Potential

In the International System of Units (SI), the electric potential is expressed in units of joules per coulomb (i.e., volts), and differences in potential energy are measured with a voltmeter.

Let us use positive charge q` to calculate the potential at point A that is distanced at r from a charge q.

Now, the force exerted by the test charge q` on the charge q placed at point A will be as such,

F= k(q.q`)/r² .….(i)

The electric field due to charge q at point A will be given as such, E= F/ q` (since the electric field is force per unit charge) 

Placing the value of F from equation 1 we get, 

E= {k(q.q`)/r2}/q` 

or, E= kq/r2 ….(2) 

Therefore, the value of the electric potential at point A can be given as, 

V= E*r …..(3) 

Now, by placing the value of E from equation 2 in equation 3 we get, 

V=( kq/r2)*r 

thus,

V = kq/r

Potential Due to Point Charge

Consider a point charge q placed at point O. Consider any point P in the field of the above charge.

Let us calculate the potential at point P due to the charge q kept at point O. Since the work done is independent of the path, we choose a convenient path, along the radial direction.

Let the distance OP = r

The electric force at P, due to q will be directed along OP, given by:

\(F=\frac{1}{4\pi \epsilon_{o}}\frac{q q_{o}}{r^{2}} \)

If the work done by moving this positive charge to dr distance is dW then,

dW = F (-dr)

\(dW=-\int F.dr\)

\(dW=-\int_{∞}^{r} F.dr\)

Hence, the total work done in bringing this charge from \(\infty\) to 'r' will be,

\(W=-\int_{∞}^{r} \frac{1}{4\pi \epsilon_{o}}\frac{q q_{o}}{r^{2}}.dr\)

\(W=-\frac{q q_{o}}{4\pi \epsilon_{o}}\int_{∞}^{r} \frac{1}{r^{2}}.dr\)

\(W=-\frac{q q_{o}}{4\pi \epsilon_{o}}\left[-\frac{1}{r}\right]_∞^r\)

\(W=\frac{1}{4\pi \epsilon_{o}}\frac{q q_{o}}{r^2}\)
Hence, from V = W/qo, electric potential is:

\( V=\frac{1}{4\pi \epsilon}_o \frac{q}{r}\)

Potential Due to Electric Dipole

Consider dipole of length ‘2a’. Let P be a point at distance r₁ from +q and r₂ from -q. Let ‘r’ be the distance of P from the centre ‘O’ of the dipole. Let θ be an angle between dipole and line OP.

The potential due to +q,

\(V_+=\frac{1}{4\pi \epsilon}_o \frac{+q}{r_1}\)

The potential due to -q,

\(V_- =\frac{1}{4\pi \epsilon}_o \frac{-q}{r_2}\)

Therefore total potential,

\(V=\frac{1}{4\pi \epsilon}_o \frac{+q}{r_1}\frac{1}{4\pi \epsilon}_o \frac{-q}{r_2}\)

\(V=\frac{q}{4\pi \epsilon}_o \left(\frac{1}{r_1}-\frac{1}{r_2}\right)\)

\(V=\frac{q}{4\pi \epsilon}_o \left(\frac{r_2-r_1}{r_1r_2}\right) \)___(1)

From ∆ , we get (r₂ – r₁) = 2a cosθ

We can also take r₂ = r₁ = r (since ‘2a’ is very small) Substituting these values in equation (1), we get

\(V=\frac{q}{4\pi \epsilon}_o \left(\frac{2a.cos\theta}{r^2}\right)\)

\(V=\frac{1}{4\pi \epsilon}_o \left(\frac{P.cos\theta}{r^2}\right)\) (∵ P=2qa)

Case 1: If the point lies along the axial line of the dipole, then θ = 0°

\(V=\frac{1}{4\pi \epsilon}_o \left(\frac{P.cos0}{r^2}\right)\)

\(V=\frac{1}{4\pi \epsilon}_o \left(\frac{P}{r^2}\right)\)

Case 2: If the point lies along the equatorial line of the dipole, then θ = 90°

\(V=\frac{1}{4\pi \epsilon}_o \left(\frac{P.cos90}{r^2}\right)\)

V = 0

Read More: Unit of Capacitance

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What is an Equipotential Surface?

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 A surface having the same electrostatic potential at every point on it, is referred to as an equipotential surface. The shape of an equipotential surface due to line charge is cylindrical and point charge is spherical.

Properties of Equipotential Surfaces

• Equipotential surfaces don't intersect each other as it gives two directions of electric field E at an intersecting point which is not possible.
• Equipotential surfaces are closely spaced in the region of strong electric field and vice-versa.
• The electric field is always normal to the equipotential surface at every point of it and directed from one equipotential surface at higher potential to the equipotential surface at lower potential.
• Work done in moving a test charge from one point to another will be zero.

The Relationship between electric field and the potential gradient is explained as:

In this, a negative sign means that the direction of the electric field is from higher potential to lower potential, i.e it is shown in the decreasing potential.

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What is Capacitor?

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The capacitor which is also referred as Condenser is two terminal electric components having the ability or capacity to store energy in the form of electric charge. The storing capacity of capacitance differs from small storage to high storage.

Capacitor

Read More: Equipotential Surfaces

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Capacitance in Physics

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The ability of a capacitor to store the energy in the form of electric charge is described as Capacitance. In similar words, capacitance is the storing ability of capacitor and the unit in which they are measured in farads.

Capacitor in Series and in Parallel

In Series

Capacitors connected in series

Both Capacitor C₁ and C₂ can easily get connected in series. When the capacitors are connected serially than the total capacitance that is C_total is less any one of the capacitor’s capacitances.

The capacitance of a capacitor connected in series is judged as:

In Parallel

Capacitor connected in parallel

Capacitors C₁ and C₂ are connected in parallel. When the capacitors are connected parallel the total capacitance that is C_total is any one of the capacitor’s capacitance

The capacitance of a capacitor connected in parallel will be explained as:

Electrical Capacitance

V ∝ Q

Or

Q = C V

C here is the proportionality constant

Parallel Plate Capacitor

The setup of dielectric and electrode to create a capacitor is explained as a parallel plate capacitor.

• The two parallel conductive plates (electrodes) are considered and in between the medium of dielectric.
• The value of capacitance depends on the material or the medium between the two plates.

A is explained as the common area between the two plates; d is the distance among the plates.

Read More: 

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Things to Remember

• Electrostatics is the branch of physics dealing with the study of interactions between stationary and slow-moving charges.
• A few examples of Electrostatic Phenomenon includes: Laser machine and photocopier, The spontaneous explosion of grain silos etc.
• The electrostatic potential is the force that is external but conservative.
• The charge depends upon the addition or removal of electrons from an atom. In case there are more protons as compared to electrons, then in this case atoms will be positively charged while in case there are more electrons than protons, the atom will be negatively charged.
•  A surface having the same electrostatic potential at every point on it, is being referred to as an equipotential surface.
• The capacitor which is also referred as Condenser is two terminal electric components having the ability or capacity to store energy in the form of electric charge.

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Previous Year Questions

1. A bar magnet is10 cm  long is kept with its north….[BITSAT 2019]
2. A conducting sphere of radius R=20cm is given a charge Q….[JIPMER 2015]
3. A metallic sphere is placed in a uniform electric field. The line of force follow the path (s) shown….[VITEEE 2018]
4. As shown in the figure, charges are placed at the vertices…. [VITEEE 2011]
5. On moving a charges , the potential difference between the points is… [VITEEE 2011]
6. Voltage rating of a parallel plate capacitor is….[JEE Main 2018]
7. In the figure shown below, the charge on the left plate of the 10μF capacitor is −30μC….[JEE Main 2019]
8. In The Figure Shown After The Switch S Is Turned from postion a to b….[JEE Main 2019]
9. the effective capacitance of the whole circuit is to…...[JEE Main 2019]
10. In the circuit shown, find C if the effective capacitance of the whole circuit is….[JEE Main 2019]
11. A solid conducting sphere, having a charge Q, is surrounded by an uncharged conducting….​.[JEE Main 2019]
12. A Plane Electromagnetic Wave Of Frequency 50 MHz travels in….[JEE Main 2019]
13. A Parallel Plate Capacitor With Square Plates Is F….[JEE Main 2019]
14. The dielectric constant of a material which when fully inserted in above capacitor, gives….[JEE Main 2019]
15. Watt per ampere is unit of…