Electric Potential and Capacitance: Structured 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.

• Formula: $W = qEd$ (for uniform field and displacement $d$)

• Example: A positive charge released in an electric field accelerates in the field's direction.

Electric Potential Energy

Electric potential energy is the energy a charge possesses due to its position in an electric field. It is analogous to gravitational potential energy.

• Gravitational Analogy: $W = mgh$ for gravity, $W = qEs$ for electric fields.

• Uniform Field: For parallel plates, $U = U_0 - qEx$

• Key Equations:

  • $F_e = qE$

  • $U = qV$

• Example: A charge moving between parallel plates loses or gains potential energy depending on its sign and direction.

Electric Potential (Voltage) Between Parallel Plates

Electric potential (voltage) is defined as potential energy per unit charge. In a uniform field (such as between parallel plates), the potential varies linearly with position.

• Formula: $V = V_0 - Ex$

• Equipotential Lines: Lines of constant potential, perpendicular to field lines.

• Example: If $V_0 = 9.0$ V at one plate and $x$ increases toward the other plate, $V$ decreases linearly.

Conservation of Energy Using Electric Potential

Energy conservation relates kinetic and potential energy for charges moving in electric fields.

• Equation: $K_a + U_a = K_b + U_b$

• Potential Energy: $U = qV$

• Example: An electron released from the negative plate of a capacitor accelerates toward the positive plate, converting potential energy to kinetic energy.

Electric Potential Energy Between Point Charges

The potential energy between two point charges depends on their separation and magnitudes.

• Force: $F = k \frac{|qQ|}{r^2}$

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

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

Electric Potential from a Point Charge

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

• Formula: $V = k \frac{Q}{r}$

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

• Example: The potential decreases with increasing distance from the charge.

Comparisons of Potential and Potential Energy

Comparing potential and potential energy at different positions helps understand their dependence on charge and distance.

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

• Potential: $V_1 = k \frac{Q}{r}$, $V_2 = k \frac{Q}{2r} = \frac{1}{2}V_1$

• Example: The potential energy can be the same for different charge-distance combinations, but the potential itself changes.

Quantity	Formula	Depends on
Electric Potential Energy	$U = k \frac{qQ}{r}$	Both charges and distance
Electric Potential	$V = k \frac{Q}{r}$	Source charge and distance