Electric Potential and Capacitance: Study Notes

Electric Potential and Capacitance

Work Done by an Electric Field

When a charge moves in the direction of an electric field, the field does work on the charge. The sign and magnitude of this work depend on the direction of motion and the charge's sign.

• Positive Work: The electric field does positive work on a positive charge moving in its direction.

• Negative Work: The field does negative work on a negative charge moving in the direction of the field.

• No Work: If the charge moves perpendicular to the field, no work is done.

Electric Potential Energy

Electric potential energy is the energy a charge possesses due to its position in an electric field. The concept is analogous to gravitational potential energy near Earth's surface.

• Gravitational Force: $\vec{F}_G = G\frac{mM}{r^2}$, $\vec{F}_G = mg$

• Electric Force: $\vec{F}_E = k\frac{|qQ|}{r^2}$, $\vec{F}_E = q\vec{E}$

• Work in Gravitational Field: $W = -\Delta U_{grav} = mgh$

• Work in Electric Field: $W = -\Delta U_{elec} = qEs$

Potential Difference Between Parallel Plates

In a uniform electric field (such as between parallel plates), the potential difference and electric potential energy can be calculated as follows:

• Work Done: $W = Fx = qEx$

• Electric Potential Energy: $U = U_0 - qEx$

• Electric Potential (Voltage): $V = \frac{U}{q}$

• Potential Difference: $V = V_0 - Ex$

Equipotential Lines

Equipotential lines are lines of equal electric potential. They are always perpendicular to electric field lines.

• Definition: Equipotential lines represent locations where the electric potential is constant.

• Properties:

  • Never intersect.

  • Moving a charge along an equipotential requires no work.

  • Electric field lines are perpendicular to equipotential lines.

Conservation of Energy Using Potential

When a battery is connected across parallel plates, the energy conversion can be analyzed using conservation of energy:

• Energy Conservation: $K_a + U_a = K_b + U_b$

• Potential Energy: $U = qV$

• Example: An electron released from the negative plate will gain kinetic energy as it moves to the positive plate, calculated by $K_f = qV$.

Electric Potential Energy Between Point Charges

The potential energy between two point charges depends on their separation:

• Force: $F = k\frac{|qQ|}{r^2}$

• Potential Energy: $U = k\frac{qQ}{r}$

• Example: For three charges arranged linearly, the total energy is the sum of pairwise potential energies.

Electric Potential from a Point Charge

The electric potential due to a point charge depends only on the distance from the charge:

• Electric Field: $E = k\frac{|Q|}{r^2}$

• Electric Potential: $V = k\frac{Q}{r}$

• Potential Energy: $U = k\frac{qQ}{r}$

Comparing Electric Potential and Potential Energy

Comparisons can be made between different positions and charges:

• Potential Energy: For charges $+q$ at $r$ and $+2q$ at $2r$, $U_1 = U_2$.

• Potential: For a charge at $r$ and at $2r$, $V_1 = k\frac{Q}{r}$, $V_2 = k\frac{Q}{2r} = \frac{1}{2}V_1$.

Summary Table: Key Equations

Quantity	Equation	Units
Electric Field (point charge)	$E = k\frac{|Q|}{r^2}$	V/m
Electric Potential (point charge)	$V = k\frac{Q}{r}$	V
Potential Energy (two charges)	$U = k\frac{qQ}{r}$	J
Work (uniform field)	$W = qEx$	J

Example Application

Suppose three charges (2.0 nC, 1.0 nC, -3.0 nC) are arranged linearly with 1.0 m separation. The total energy stored is the sum of the potential energies for each pair:

• Calculate $U_{12}$, $U_{13}$, $U_{23}$ using $U = k\frac{qQ}{r}$.

• Add the results for total energy.

Additional info: These notes expand on the brief points and diagrams in the original slides, providing full definitions, equations, and context for each concept. The summary table is inferred for clarity and completeness.