#3 Slide Notes

The Electric Potential of a Charged Sphere

• Outside a Uniformly Charged Sphere:
  • The electric potential is identical to that of a point charge $Q$ located at the center of the sphere.
• Potential at the Surface of the Sphere:
  • For a sphere of radius $R$, which is charged to a potential, has total charge: $V = k\frac{Q}{R}$
    where $k$ is Coulomb's constant.
• Dependence: The potential of the sphere decreases inversely with the distance from the charge.

The Electric Potential Inside a Parallel-Plate Capacitor

• Electric Field Inside the Capacitor:
  • The electric field $E$ inside the parallel-plate capacitor is uniform.
• Electric Potential Energy of a Charge $q$:
  • The electric potential energy $U$ of a charge $q$ in the uniform electric field of a parallel-plate capacitor is given by: $U = qV$
• Electric Potential $V$ Inside the Capacitor:
  • The potential inside the capacitor varies with distance $s$ from the negative electrode.

Graphical Representations of Electric Potential in Capacitors

• Equipotential Surfaces:
  • These are mathematical surfaces where the electric potential $V$ remains constant at every point.
  • For capacitors, these surfaces are planes that are parallel to the capacitor plates.

Equipotentials and Electric Field Vectors in a Parallel-Plate Capacitor

• Relationship Between Electric Fields and Equipotential Surfaces:
  • The electric field vectors are always perpendicular to the equipotential surfaces.
  • The electric field points in the direction of decreasing potential, interpreted as pointing "downhill" on a graph or map representing the electric potential.

The Electric Potential of Many Charges

• Calculation of Electric Potential at a Point in Space:
  • The electric potential $V$ at a point in space due to multiple source charges $q<em>1, q</em>2,\dots$ is the sum of the potentials due to each individual charge.
• Principle of Superposition:
  • Similar to electric fields, the electric potential obeys the principle of superposition.
• Potential of an Electric Dipole:
  • The potential of an electric dipole is the sum of the potentials due to the positive and negative charges that constitute the dipole.

The Potential of a Continuous Distribution of Charge

• Procedure for Calculating Potential:
  • This procedure is similar to calculating the electric field of a continuous charge distribution but is easier due to the scalar nature of potential.
• Steps Involved:
  1. Divide the total charge $Q$ into small pieces of charge.
  2. Use shapes where the potential $V$ can be easily determined.
  3. Calculate the potential due to each segment of charge.
  4. Apply the superposition principle to obtain the total potential.

The Potential of a Ring of Charge

• Ring of Charge Characteristics:
  • A thin, uniformly charged ring of radius $R$ with total charge $Q$.
• Finding the Potential:
  • The potential at a distance $z$ on the axis of the ring can be calculated using appropriate formulas specific to the geometry of the ring.

The Potential of a Charged Disk

• Description:
  • A thin, plastic disk of radius $R$ is uniformly coated with charge until it reaches a total charge of $Q$.
• Calculations Required:
  1. The potential at distance $z$ along the axis of the disk.
  2. The potential energy when an electron is at a distance of 1.00 cm from a 3.50-cm-diameter disk, charged to 5 µC.

Calculation Details for Potential of Charged Disk

• Integral for Potential at Distance $z$ of the Disk: $\frac{1}{4\pi\varepsilon_0} \times \frac{Q}{R^2} \times \frac{z}{R^2 + z^2}$
• Potential Energy Calculation:
  • For potential energy $U$ of an electron at a distance $d$: $U = qV$
    Where:
  • $qV = -4.77 \times 10^{-16} J.$

Analysis of Electric Potential Zones

• Question Consideration:
  • At which point or points is the electric potential zero amongst specific reference points labeled A, B, C, D?
  • Note: There may be more than one point with zero electric potential