Chapter 16 Electrical Energy and capacitance

16.1 Potential Difference and Electric Potential

• The electrostatic force is a conservative force. This means that the work done on an object depends only on the initial and final positions of the object and not on the path.

• There is an electrostatic potential energy (EPE) analogous to the gravitational potential energy (GPE).

F = F =

= q_o = m

• When a small positive charge moves from A to B (Figure 16.1) under the influence of the electric force exerted on it, qE, the work done on the charge by the electric force is

W_AB = Fd = qEd

• The work done by a conservative force equals the negative of the change in potential energy, DPE.

DPE = -W_AB = -qEd (16.1)

� Definition of (Electric) Potential Difference �

The potential difference between points A and B, DV V_B – V_A, is defined as the change in potential energy of a charge, q, moved from A to B, divided by the charge.

DV V_B – V_A = (16.2)

The unit of Potential difference V is V (volt) = joule/coulomb. The electric potential is a scalar quantity.

• The (electric) potential difference between two points, A and B, in a uniform electric field, E, is

DV V_B – V_A = = -Ed (16.4)

where d is the distance between A and B.

• Figure 16.2

• Examples

16.2 Electrical Potential and Potential Energy Due to Point Charges

• The electric potential created by a point charge, q, at any distance, r, from the charge is given by

V = k (16.5)

• The total electric potential at some point P due to several point charges is the algebraic sum of the electric potentials due to the individual charges. (Superposition Principle)

€ Electric Potential Energy of Two Charged Particles €

The electric potential energy of a pair of point charges separated by distance r is

PE = k (16.6)

• Computer generated image on page 526

• Examples

16.3 Potentials and Charged Conductors

• The relation between work and electric potential:

W_AB = -DPE

DPE = q (V_B - V_A)

W_AB = - q (V_B - V_A)

• No work is required to move a charge between two points that are at the same electric potential (equipotential points).

• Every point on the surface of a charged conductor in electrostatic equilibrium is at the same potential. Furthermore, the potential is constant everywhere inside the conductor and equals its value on the surface. Consequently, no work is required to move a charge from the interior of a charged conductor to its surface. (Figure 16.7)

<The Electron Volt>

• The electron volt is defined as the energy that an electron gains when accelerated through a potential difference of 1 V.

1eV = (1.6 x 10^-19 C) x (1 V) = 1.6 x 10^-19 J

16.4 Equipotential Surfaces

• A surface on which all points are at the same potential is called an equipotential surface. No work is required to move a charge at constant speed on an equipotential surface.

• The electric field at every point of an equipotential surface is perpendicular to the surface.

• Figure 16.8

16.5 Applications (Reading Assignment)

16.6 The Definition of Capacitance

• A capacitor consists of two conductors of any shape placed near one another without touching. (Figure 16.11) It is common practice to fill the region between the conductors or plates with an electrically insulating material called a dielectric.

• A capacitor stores electric charge. Each capacitor place carries a charge of the same magnitude, one positive and the other negative. The electric potential of the positive plate exceeds that of the negative plate by an amount V.

� The Relation Between Charge and Potential Difference for a Capacitor �

The magnitude Q of the charge on each plate of a capacitor is directly proportional to the magnitude V of the potential difference between the plates:

C = Q / DV; Q = CV (16.5)

where C is the capacitance.

Unit of Capacitance: coulomb/volt = F (farads)

• 1F = 10^-6 F; 1 pF = 10^-12 F

• Examples

16.7 The Parallel-Plate Capacitor

• The capacitance of a capacitor is affected by the geometry of the plates and the dielectric constant of the material between them.

• The capacitance C of a parallel capacitor filled with air is:

C = (16.10)

where A is the area of one of the plates, d is the distance of separation of the plates, and is a constant called the permittivity of free space, with the value = 8.85 x 10^-12 C²/N-m².

k =

• Examples

• Symbol

16.8 Combinations of Capacitors

<Parallel Combination>

• V = V₁ = V₂

• Q = Q₁ + Q₂

• C_eq = C₁ + C₂

<Series Combination>

• V = V₁ + V₂

• Q = Q₁ = Q₂

• 1/C_eq = 1/C₁ + 1/C₂

16.9 Energy Stored in a Charged Capacitor

• A capacitor is a device for storing charge. Alternatively, it is possible to view the capacitor as a device for storing energy.

• The energy stored in a capacitor is

Energy Stored = QV = CV² = (16.18)

16.10 Capacitors with Dielectrics (Reading Assignment)