Chapter 22: Electric Potential – Physics II Study Notes

Electric Potential & Potential Energy

Introduction to Electric Potential and Potential Energy

Electric potential and potential energy are fundamental concepts in electrostatics, describing how energy is stored and transferred in systems of charges. These concepts are analogous to gravitational potential energy in mechanics.

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

• Electric Potential (V): The potential energy per unit charge at a point in space.

• Analogy: Just as gravitational potential energy is stored in a gravitational field, electric potential energy is stored in an electric field.

Formula for Electric Potential Energy in a Uniform Field:

• $U = q E x$

• Where q is the charge, E is the electric field strength, and x is the position.

Formula for Gravitational Potential Energy:

• $U_g = m g y$

• Where m is mass, g is gravitational acceleration, and y is height.

Potential Energy of Point Charges

Electrostatic Potential Energy for Point Charges

When dealing with multiple point charges, the potential energy depends on the configuration and separation of the charges.

• For two point charges:

• $U = k_e \frac{q_1 q_2}{r}$

• Where k_e is Coulomb's constant, q_1 and q_2 are the charges, and r is the separation.

• For multiple charges:

• $U_{\text{total}} = \sum_{i

• Sum over all unique pairs of charges.

Key Points:

• Negative potential energy indicates an attractive force.

• Positive potential energy indicates a repulsive force.

• Potential energy is zero when charges are infinitely far apart.

Example: Four identical charges at the corners of a rectangle – calculate the total potential energy by summing the energy for each unique pair.

Electric Potential

Definition and Calculation

Electric potential is a scalar quantity that represents the potential energy per unit charge at a point in an electric field.

• $V = \frac{U}{q}$

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

• Work done by the field: $W = -q \Delta V$

Electric Potential due to a Point Charge:

• $V = k_e \frac{Q}{r}$

• Where Q is the source charge and r is the distance from the charge.

Example: Ratio of potentials at different points from a charge: $V_B/V_A = r_A/r_B$

Electric Potential in a Uniform Electric Field

Relationship Between Field and Potential

In a uniform electric field, the potential difference between two points is related to the field strength and the distance between the points.

• $\Delta V = -E \Delta x$

• Work done by the field: $W = q E \Delta x$

Example: Calculating the change in potential energy as a charge moves in a uniform field.

Sources of Electric Potential

Common Sources and Potential Differences

Electric potential differences are created by various sources, including batteries, static electricity, and biological systems.

Source of Potential	Approximate Potential
Brain activity (EEG)	10–100 μV
Cells in human body	100 mV
Batteries	1–10 V
Household electricity	100 V
Static electricity	10,000 V
Transmission lines	100 kV

Key Point: Only potential differences are physically meaningful and measurable.

Equipotential Surfaces

Definition and Properties

Equipotential surfaces are regions where the electric potential is constant. No work is required to move a charge along an equipotential surface.

• Equipotential surfaces are always perpendicular to electric field lines.

• Work done moving a charge along an equipotential: $W = 0$

Example: Calculating work done and potential differences between equipotential surfaces.

Equipotentials & Field Lines

Relationship Between Equipotentials and Field Lines

Electric field lines indicate the direction of force on a positive charge, while equipotential surfaces show regions of constant potential.

• Equipotential surfaces intersect field lines at right angles.

• No work is done by the field when moving along an equipotential.

Connecting Electric Field & Electric Potential

Mathematical Relationship

The electric field is related to the spatial rate of change of electric potential.

• $\vec{E} = -\nabla V$

• The electric field points in the direction of greatest decrease in potential.

• Field lines are always perpendicular to equipotential surfaces.

Example: Determining the direction of the electric field and potential at the midpoint between two opposite charges.

Other Electric Potential Expressions

Potential Due to Various Charge Distributions

Electric potential can be calculated for different charge distributions, such as rings or disks.

• Point charge: $V = k_e \frac{Q}{r}$

• Ring of charge: $V = k_e \frac{Q}{\sqrt{a^2 + x^2}}$

• Disk of charge: $V = 2\pi k_e \sigma \left[ \sqrt{R^2 + x^2} - x \right]$

Example: Calculating the potential at a point due to a ring or disk of charge.

Summary Table: Key Equations

Concept	Equation
Electric Potential (Point Charge)	$V = k_e \frac{Q}{r}$
Potential Energy (Two Charges)	$U = k_e \frac{q_1 q_2}{r}$
Potential Difference (Uniform Field)	$\Delta V = -E \Delta x$
Work Done by Field	$W = -q \Delta V$
Electric Field from Potential	$\vec{E} = -\nabla V$

Examples & Applications

• Calculating the change in potential energy as a charge moves in a field.

• Finding the electric potential at a point due to multiple charges.

• Determining the work required to assemble a configuration of charges.

• Analyzing equipotential surfaces and their relationship to field lines.

Additional info: These notes expand on the original slides by providing full definitions, formulas, and context for each concept, ensuring a self-contained study guide for exam preparation.