Chapter 17: Electric Potential and Capacitance – Study Notes

Electric Potential Energy and Potential Difference

Conservative Forces and Potential Energy

Electric force, like gravitational force, is a conservative force. This means the work done by the force depends only on the initial and final positions, not the path taken. The concept of electric potential energy (PE) is analogous to gravitational potential energy.

• Electric Potential Energy (PE): The energy a charge has due to its position in an electric field.

• Electric Potential (V): The potential energy per unit charge, defined as $V = \frac{PE}{q}$.

• Potential Difference (ΔV): The work done to move a charge between two points, $\Delta V = \frac{\Delta PE}{q}$.

• The unit of electric potential is the volt (V), where $1\,\text{V} = 1\,\text{J}/\text{C}$.

Example: Setting the potential at ground to zero ($V=0$) is a common reference.

Relationship Between Electric Potential and Electric Field

The change in electric potential energy is related to the work done by the electric force:

• $\Delta PE = qEd$ for a uniform electric field.

• Whether energy increases or decreases depends on the sign of $q$.

• For gravitational potential: $V = \frac{PE}{m} = gy$; for electric: $V = \frac{PE}{q}$.

Electric Potential: Definitions and Units

Electric Potential and Potential Difference

Electric potential is defined as the energy per unit charge. The potential difference between two points is:

• $\Delta V = V_2 - V_1 = Ed$ (for uniform fields)

• Once $\Delta V$ is known, the change in potential energy is $\Delta PE = q\Delta V$.

• Unit: Volt ($V$), where $1\,\text{V} = 1\,\text{J}/\text{C}$.

Electric and Gravitational Potential Energy

Both electric and gravitational potential energies depend on the sign of the charge or mass:

• For positive $q$, $PE_a > PE_b$ if $V_a > V_b$.

• For negative $q$, $PE_a < PE_b$ if $V_a > V_b$.

Electric Field and Potential Difference Between Plates

Direction of Electric Field and Forces

In a parallel plate setup, the electric field points from the plate with higher potential to lower potential. The force on a positive charge is in the direction of the field, while for a negative charge, it is opposite.

• Convenient to set $V=0$ at ground.

• Potential energy depends on the sign of the charge.

Work, Energy, and Conservation

Conservation of Energy in Electric Fields

When a charge moves in an electric field, its kinetic energy increases as its potential energy decreases:

• $\Delta KE = -\Delta PE = W_{AB}$

• For an electron released from rest, the final velocity can be calculated using energy conservation.

Relation Between Electric Potential and Electric Field

Uniform Electric Field

For a uniform electric field, the potential difference is given by:

• $\Delta V = Ed$

• Example: Voltage across a cell membrane with thickness $d$ and field $E$.

The Electron Volt: A Unit of Energy

Definition and Application

The electron volt (eV) is the energy gained by an electron moving through a potential difference of one volt:

• $1\,\text{eV} = 1.6 \times 10^{-19}\,\text{J}$

• Useful for atomic and subatomic processes.

Electric Potential Due to Point Charges

Potential Around a Point Charge

The electric potential at a distance $r$ from a point charge $Q$ is:

• $V = \frac{kQ}{r}$

• For $Q > 0$, $V > 0$; for $Q < 0$, $V < 0$.

• $V = 0$ when $r = \infty$.

Potential Due to Multiple Charges

The total potential at a point due to several charges is the algebraic sum of the potentials from each charge:

• $V = \sum \frac{kQ_i}{r_i}$

Equipotential Lines and Surfaces

Definition and Properties

Equipotential surfaces are surfaces where the electric potential is constant. They are always perpendicular to the electric field lines.

• No work is done when moving a charge along an equipotential surface.

• Metal surfaces are equipotential surfaces.

Capacitance and Capacitors

Definition and Parallel Plate Capacitor

A capacitor is a device that stores electric charge and energy. It consists of two conducting plates separated by an insulating gap.

• Capacitance (C): $C = \frac{Q}{V}$

• Unit: Farad ($F$), $1\,F = 1\,C/V$

• Parallel plate capacitor: $C = \epsilon_0 \frac{A}{d}$

Dielectrics

The gap between capacitor plates can be filled with a dielectric material, which increases capacitance by reducing the electric field.

• Dielectric constant (κ): Dimensionless factor describing the effect of the material.

• Capacitance with dielectric: $C = \kappa \epsilon_0 \frac{A}{d}$

• Electric field is reduced by a factor of $\kappa$.

Material	Dielectric constant κ
Vacuum	1.00000
Air	1.00059
Bakelite	4.9
Fused quartz	3.78
Neoprene rubber	6.7
Nylon	3.4
Paper	3.7
Polystyrene	2.56
Pyrex glass	5.6
Silicon oil	2.5
Strontium titanate	233
Teflon	2.1
Water	80

Capacitors in Series and Parallel

Series and Parallel Configurations

Capacitors can be connected in series or parallel, affecting the total capacitance:

• Series: $\frac{1}{C_s} = \frac{1}{C_1} + \frac{1}{C_2} + \cdots$

• Parallel: $C_p = C_1 + C_2 + \cdots$

Energy Storage in Capacitors

Energy Stored and Applications

The energy stored in a capacitor is given by:

• $E = \frac{1}{2}CV^2$

• Capacitors are used in devices such as heart defibrillators, where stored energy is released rapidly.

Summary of Key Equations

• Electric potential: $V = \frac{PE}{q}$

• Potential difference: $\Delta V = Ed$ (uniform field)

• Potential due to point charge: $V = \frac{kQ}{r}$

• Capacitance: $C = \frac{Q}{V}$

• Parallel plate capacitor: $C = \epsilon_0 \frac{A}{d}$

• Capacitance with dielectric: $C = \kappa \epsilon_0 \frac{A}{d}$

• Energy stored: $E = \frac{1}{2}CV^2$

Additional info: Digital electronics and binary numbers are briefly mentioned as applications of electric potential in signal processing, but are not covered in detail in these notes.