Chapter 4: Electric Potential and Capacitance

It deals with the potential associated with electric charges and the capacitance associated with capacitors.

Electrical Potential Energy

We is the work done by the electric force, then the change in the charge’s electrical potential energy is defined by:

• Ue = electrical potential energy

• We = work done by electric force

Electrical Potential Energy from a point charge

Electrical potential energy required to move along the field lines surrounding a point charge is given by:

• q1 and q2 = charges

• e0 = permeability of free space

• Ue = electrical potential energy

• r = distance

Electric Potential

Electric potential is the electric potential energy per unit of charge at a point in an electric field, measured in volts (V). It's the work done per unit charge in bringing a test charge from infinity to that point.

***V = U/q***

Electric Potential Energy from a point charge

Consider the electric field created by a point source charge Q. If a charge moves from a distance rA to a distance rB from Q, then the change in the potential energy is:

• Ub and Ua = electrical potential energies for a and b

• ra and rb = distances for a and b

• e0 = permeability of free space

Equipotential Surface

An equipotential surface is a surface in a region of space where every point on the surface is at the same potential. In other words, no work is required to move a charge along an equipotential surface. Equipotential surfaces are perpendicular to electric field lines and can be used to visualize the electric field in a given region.

Addition of electric potential

V = kQ/r

• V = electric potential energy

• q = point charge

• r = distance between any point around the charge to the point charge

• k = Coulomb constant; k = 9.0 × 109 N

Equipotential Curve

Equipotential curves are curves of constant elevation. If you walk along any of the contour lines and you neither ascend nor descend, then the curve is known as the equipotential curve.

Equipotential Map

A drawing of several equipotential curves at various values of the potential for a charge distribution is called an equipotential map.

Capacitor

Two conductors, separated by some distance carry equal but opposite charges +Q and -Q. The pair comprises a system called a capacitor.

Parallel-Plate Capacitor

The capacitor is in the form of parallel metal plates or sheets.

Capacitance

The capacitance measures the capacity for holding charge.

***C = κε₀A/d (k = dielectric constant)***

Fringing fields

Fringing fields extend beyond conductor or magnetic material edges. They weaken as the distance from the edge increases. They're important in device design but can cause interference and affect performance.

energy stored in capacitor

The energy stored in a capacitor can be calculated using the formula

***Uc = ½QV = ½CV²***

where U is the energy stored in joules, C is the capacitance of the capacitor in farads and V is the voltage across the capacitor in volts.

Dielectric

To keep the plates of the capacitor apart they are filled with dielectric which increases the capacitance of the capacitor.

The amount of work done by a uniform electric field

W = q E d

• W = work done

• q = charge

• E = electric field

• d = distance

A dielectric always _______ the capacitance of a capacitor

increases

the __________ of the capacitance of a collection of capacitors in series is found by adding the reciprocals of the individual capacitances

reciprocal

Collection of capacitors are said to be in ______ if they all share the same charge magnitude

series

The equivalent capacitance of a collection of capacitors in parallel is found by _______ the individual capacitances

adding

The equivalent capacitance of a collection of capacitors in parallel is found by _______ the individual capacitances

parallel

1 C/V = ?

1 Farad

Ratio of charge to potential difference \n (C=Q/∆V)

capacitance

Magnitude of the potential difference between two plates of a distance d

***∆V = -Ed***

How does the electric field relate to electric potential difference?

***E = -dV/dr***

1 volt = ?

1 J/C

Wₑ = Fₑ\*distance\*cosθ = qE\*distance; ∆U =?

\-qEd\*cosθ

When a charge moves in an electric field, unless its displacement is always perpendicular to the field, the electric force does _______ on the charge.

work

Factors on which capacitance of a parallel plate capacitor depend on?

1\. Area of the plates (C ∝ A) \n 2. Distance between the plates ( C ∝ 1/d) \n 3. Permittivity of medium ( C ∝ ε0)

What is the dimensional formula of capacitance?

M-1L-2T4A2

Potential energy of a dipole in a uniform electric field?

U = -pEcosθ

How to solve for capacitance?

C = q/V

What factors does capacitance depend on?

1\. Size and shape of conductor \n 2. Nature (permittivity)

How to find total capacitance in a series combination?

1/Cs = 1/C1 + 1/C2 + 1/C3

How to find total capacitance in a parallel combination?

Cp = C1 +C2 + C3