Electric Potential and Potential Energy: Study Notes (Giancoli Ch. 23)

Electric Potential

Work and Conservative Forces

The concept of work and conservative forces is fundamental in understanding electric potential. The work done by a possibly varying force along a generic trajectory is given by:

• Work (W): $W = \int_{a}^{b} \vec{F} \cdot d\vec{r}$ (line integral, scalar product)

• If $\vec{F} \cdot d\vec{r} = 0$ everywhere, then $W = 0$.

• Conservative Force: A force is called conservative if the work done by it depends only on the initial and final positions of the object, not on the path. For conservative forces, we define the notion of potential energy $U$.

• Potential energy change: $\Delta U = -W$ (work done by conservative force lowers potential energy)

• Examples: Gravity, electrostatic force

Electric Potential Energy and Electric Potential

Electrostatic Forces and Potential

Electrostatic forces, like gravitational forces, are conservative. For a given electrostatic field, using a test charge $q_0$, we define:

• Electric Field (\vec{E}): $\vec{E} = \frac{\vec{F}}{q_0}$

• Electric Potential (V): $V = \frac{U}{q_0}$ (units: Volts, V)

• Relation: $\vec{E} = -\nabla V$

• Equipotential surfaces: Surfaces where the electric potential is constant. The electric field always points in the direction of the greatest decrease of electric potential.

Conservation of Energy in Electric Fields

Motion of Charged Particles

When a charged particle moves in a uniform electric field, its kinetic and potential energies change according to the conservation of energy principle:

• Conservation of Energy: $\text{kinetic energy} + \text{electric potential energy} = \text{constant}$

• For a particle of charge $q$ and mass $m$ moving from $x_a$ to $x_b$ in a uniform field $E$:

$\frac{1}{2} m v_a^2 + q E x_a = \frac{1}{2} m v_b^2 + q E x_b$

Solving for speed:

$v_b^2 = v_a^2 + \frac{2qE}{m} (x_a - x_b)$

Superposition Principle for Potentials

Adding Electric Potentials

The superposition principle states that electric fields add up like vectors, and electric potentials add up like scalars:

• Electric Potential at a Point: $V = \frac{1}{4\pi\epsilon_0} \left( \frac{q_1}{r_1} + \frac{q_2}{r_2} + \frac{q_3}{r_3} \right)$

• Potentials due to multiple charges are summed algebraically.

Work Done by Electric Forces

Moving Charges in Electric Fields

To move a charge in an electric field, work must be done against the field. For two fixed point charges $Q_1$ and $Q_2$, the work to move a charge $q$ from point $a$ to point $b$ is:

• $W_{ab} = q (V_b - V_a)$

• Where $V_a$ and $V_b$ are the electric potentials at points $a$ and $b$ due to $Q_1$ and $Q_2$.

Electric Fields in Biological Systems

Electric Fields Used by Fish

Some species of fish use electric fields for detection and communication:

• A charge separation in the body produces electric fields of a dipole type.

• The field is monitored by receptors along the body.

• The field is distorted by the presence of a conducting object, allowing the fish to "see" using electric fields.

Electric Potential Energy of Interaction

Work to Bring Two Charges Together

The work required to bring two charges $Q_1$ and $Q_2$ to a distance $r$ from one another is:

• $W_{\infty \rightarrow r} = \frac{Q_1 Q_2}{4\pi\epsilon_0 r}$

• This is their electric potential energy of interaction: $U_{el} = \frac{Q_1 Q_2}{4\pi\epsilon_0 r}$

• Alternatively: $U_{st} = Q_1 V_2 = Q_2 V_1$

Application: Nuclear Alpha-Particle and Potential Energy

Minimum Approach Distance

When a nuclear alpha-particle (helium-4 nucleus) is shot towards an aluminum target, the minimum distance it can approach is determined by conservation of energy:

• Initial kinetic energy is converted to electric potential energy as the alpha-particle approaches the nucleus.

• $\frac{1}{2} m v_0^2 = \frac{q_1 q_2}{4\pi\epsilon_0 d}$

• Solving for $d$ gives the minimum approach distance.