Chapter : 2

What Are Electrostatic Potential and Capacitance ?

Electric Potential

Electric potential V at any point in a region of electric field is the minimum work done W in carrying a unit positive charge q0 from infinity to that point.
V =(w/q0)

• Electrostatic potential at a point is a scalar quantity.

• SI unit of electric potential is volt (V).

• Potential is a relative parameter as it depends on the frame of reference.

Potential Difference

Potential difference between two points A and B is defined as the amount of work done in moving a unit positive test charge from one point to another.

VB–VA = ΔV =(WAB/q)

• Potential difference does not depend on frame of reference, hence it is an absolute quantity.

Electric Potential Due To A Point Charge

Electric potential difference at a point is the difference between the potential at the point and the potential at infinity (reference point).

electricpotentialc

$$V=\frac{1}{4\pi\epsilon_0}.\frac{\text{q}}{\text{r}}$$

• A positive charge produces a positive electrostatic potential and a negative charge produces a negative electrostatic potential.

• Electrostatic potential is zero at infinity.

Electric Potential Due To A Dipole

Electric potential at a point P due to an electric dipole is V =(1/4πε0).(p cosθ/r2)

• When point P lies on the axial line of the dipole i.e., θ = 0°

V =(1/4πε0).(p/r2)

• When point P lies on the equatorial line of the dipole, i.e., θ = 90°

V = 0

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• Potential due to a point charge varies inversely as the distance from the charge i.e., V ∝ (1/r) whereas potential due to dipole falls off more rapidly, i.e., V ∝ (1/r2)

Electric Potential Due To A System Of Charges

Electric potential at a point due to a system of charges is equal to the algebraic sum of the potentials due to the individual charges.

$$\text{V}=\text{V}_1+\text{V}_2+\text{V}_3+….+\text{V}_\text{n}=\bigg(\frac{\text{q}_1}{\text{r}_1}+\frac{\text{q}_2}{\text{r}_2}+\frac{\text{q}_3}{\text{r}_3}+…….+\frac{\text{q}_\text{n}}{\text{r}_\text{n}}\bigg)\text{or} \space\text{V}=\frac{1}{4\pi\epsilon_0}\sum_{i=1}^n\frac{\text{q}_\text{i}}{\text{r}_\text{i}}$$

Equipotential Surfaces

An equipotential surface is that surface at every point of which electric potential is the same.

Relation between E and V

Magnitude of electric field is given by the change in the magnitude of potential per unit displacement normal to the equipotential surface at the point.

E = –(∂V/∂r)

Negative sign shows that the electric field is in the direction of decreasing potential.

• Electric field lines are always perpendicular to an equipotential surface.

• Work done in moving an electric charge from one point to another on an equipotential surface is zero.

• Two equipotential surfaces can never intersect one another.

Electrical Potential Energy Of A System Of Two Point Charges

Electric potential energy of a system of two point charge q1 and q2 is U =(1/4πε0)(q1q2/r12)
where r12 is the distance between q1 and q2

• The SI unit of potential energy is joule (J).

Electrical Potential Energy Due To An Electric Dipole In An Electrostatic Field

Potential energy of a dipole in an external field is

where θ is the angle between external electric field (E) and electric dipole moment (p).

• Potential energy is maximum when θ = 180°, ∴ Umax = pE
• Potential energy is minimum when θ = 0°, ∴ Umin = -pE

• The electric dipole is in unstable equilibrium when it is oriented anti-parallel to the electric field i.e., when θ = 180°.

• The electric dipole is in stable equilibrium when it is aligned parallel to the electric field i.e., q = 0°.

Conductors And Insulators

The substances which can easily allow elecricity to pass through them are called conductors. The substances which do not allow electricity to pass through them are called insulators or dielectrics.

• Mentals, humans, animal bodies and earth are conductors.

Free Charges And Bound Charges Inside A Conductor

Conductors have a large number of free charge carriers that are free to move inside the material. In constrast to conductors, insulators or dielectrics have charge carriers that are attached to specific atoms and molecules. These charges are called bound charges. These charges however, can be displaced within an
atom or a molecule.

• Inside a conductor electric field, E = 0, therefore, net charge inside a conductor is zero.

Dielectrics And Electric Polarisation

Dielectrics are non-conducting substances which have negligible number of charge carriers.

• Inside a conductor electric field,

$$\vec{\text{E}}=0,$$

therefore, net charge inside a conductor is zero.

Dielectrics And Electric Polarisation

Dielectrics are non-conducting substances which have negligible number of charge carriers.

Dielectrics

  • Polar Dielectrics
  • Polar dielectrics are made up of polar molecules. Each polar molecule has a permanent dipole moment. Examples of polar dielectrics are water (H2O) and HCl.
  • Non-polar Dielectrics
  • Non-polar dielectrics are mode up of non-polar molecules. Each non-polar molecule has zero dipole moment.
    Examples of non-polar dielectrics are Oxygen (O2) and hydrogen (H2).

Polarisation is defined as the dipole moment per unit volume. For linear isotropic dielectrics,

$$\vec{\text{P}}= χ_e \vec{\text{E}},$$

$$|\text{where}\space χ_e$$

is a constant characteristic of the dielectric and is called electric susceptibility of the dielectric medium.

Capacitors And Capacitance

A capacitor is a device that stores electric charge.

Capacitance of a capacitor is the ratio of charge (Q) given and the potential (V) to which it is raised.

C = (Q/V)

Capacitance of a spherical conductor is C = 4πε0R, where R is the radius of the spherical conductor.

• The SI unit of capacitance is farad (F).

• Capacitance is a scalar quantity.

Combination Of Capacitors In Series And In Parallel

Capacitors In Series

capacitors

• Same charge flows through each capacitor.

• Different potential difference exists across each capacitor if C1 ≠ C2 ≠ C3, such that, V = V1 + V2 + V3 or  (1/Cs)= (1/C1)+(1/C2) (1/C3)

• For n capacitors connected in series, total capacitance would be

$$\frac{1}{\text{C}_\text{S}}=\sum_\text{\text{i=1}}^{\text{i=n}}\frac{1}{\text{C}_\text{i}}$$

Capacitors in Parallel

• Different charge flows across each capacitor if C1 ≠ C2 ≠ C3, such that,
q = q1 + q2 + q3 or Cp = C1 + C2 + C3

• For n capacitors connected in parallel, total capacitance would be

$$\frac{1}{\text{C}_\text{p}}=\sum_\text{\text{i=1}}^{\text{i=n}}\text{C}_\text{i}$$

Capacitance Of A Parallel Plate Capacit Or With And Without Dielectric Medium Between The Plates

When air is betwe en the plates

• Capacitance of a parallel plate capacitor is given by, C0 =(ε0A/d),
where ε0 is the permittivity of free space, A is the area of plates and d is the separation between the plates.
• The product ε0k is called the permittivity of the medium and is denoted by ε.
∴ ε = ε0k or k =(ε/ε0)=(C/C0)

When dielectric is between the plates

• Capacitance of a parallel plate capacitor with a dielectric between the plates is given by, C=(kε0A/d),

where k is the dielectric constant.

• The capacitance of a parallel plate capacitor increases when dielectric is introduced between its plates.

Energy Stored In A Capacitor

Work done in charging a capacitor gets stored in the capacitor in the form of its electric potential energy and it is given by U =(1/2)CV2=(1/2)QV=(1/2)(Q2/C),

where C is the capacity of the capacitor.

• When a capacitor is charged with the help of a battery, the potential energy of the capacitor is obtained at the cost of chemical energy stored in the battery.