7.3 Calculations of Electric Potential

Electric potential is a crucial concept in electrostatics, measuring the potential energy per unit charge at a point in an electric field. It's calculated using the formula V = kq/r for point charges, where k is Coulomb's constant and r is the distance from the charge.

Understanding electric potential is key to grasping how charges interact and move in electric fields. It's a scalar quantity, meaning we can add potentials from multiple charges using the superposition principle. This concept extends to continuous charge distributions and complex systems.

Electric Potential

Electric potential from point charges

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• Electric potential ($V$) measures potential energy per unit charge at a point in an electric field
• For a point charge ($q$), electric potential at distance $r$ is $V = \frac{kq}{r}$
  • $k$ is Coulomb's constant: $k = 8.99 \times 10^9 \frac{N \cdot m^2}{C^2}$
  • Potential is measured in volts (V), where 1 V = 1 J/C (joule per coulomb)
• Positive charges have positive potential, negative charges have negative potential
• Electric potential decreases as distance from point charge increases (inverse relationship)
  • Example: Potential of a proton is higher near its surface than far away in space
• Electric potential is related to potential energy in electrostatics

Total potential of multiple charges

• Electric potential is a scalar quantity, can be added or subtracted
• Find total electric potential at a point using superposition principle
  • Calculate electric potential from each point charge individually
  • Add individual electric potentials to get total electric potential at the point
• Total electric potential at point $P$ due to $n$ point charges is $V_P = \sum_{i=1}^{n} \frac{kq_i}{r_i}$
  • $q_i$ is magnitude of each point charge
  • $r_i$ is distance from each point charge to point $P$
  • Example: Two positive charges of 1 C each, 1 m apart, potential at midpoint is sum of potentials from each charge

Concept of electric dipoles

• Electric dipole has two equal and opposite point charges separated by small distance
• Dipole moment ($\vec{p}$) is vector characterizing strength and orientation of dipole
  • Magnitude of dipole moment is $|\vec{p}| = qd$
    • $q$ is magnitude of each point charge
    • $d$ is distance between charges
  • Dipole moment points from negative to positive charge
• Electric potential of dipole depends on distance and angle from dipole
  • At point along dipole axis, electric potential is $V = \frac{kp\cos\theta}{r^2}$
    • $p$ is magnitude of dipole moment
    • $\theta$ is angle between dipole moment and line from dipole center to point
    • $r$ is distance from dipole center to point
  • Example: Water molecule is an electric dipole, with positive and negative ends

Continuous Charge Distributions and Superposition

Potential from continuous charge distributions

• Continuous charge distributions include line, surface, and volume charges
• Calculate electric potential from continuous charge distribution by integrating potential contributions from infinitesimal charge elements
• For line charge with linear charge density $\lambda$, electric potential at point $P$ is $V_P = \int \frac{k\lambda dl}{r}$
  • $dl$ is infinitesimal line element
  • $r$ is distance from line element to point $P$
  • Example: Uniformly charged rod
• For surface charge with surface charge density $\sigma$, electric potential at point $P$ is $V_P = \int \frac{k\sigma dA}{r}$
  • $dA$ is infinitesimal surface element
  • $r$ is distance from surface element to point $P$
  • Example: Uniformly charged sheet or plane
• For volume charge with volume charge density $\rho$, electric potential at point $P$ is $V_P = \int \frac{k\rho dV}{r}$
  • $dV$ is infinitesimal volume element
  • $r$ is distance from volume element to point $P$
  • Example: Uniformly charged sphere or cube

Superposition in complex potential problems

• Superposition principle: total electric potential at a point from multiple charge distributions is sum of individual electric potentials
• To solve complex electric potential problems:
  1. Identify all charge distributions in the system
  2. Calculate electric potential from each charge distribution individually
  3. Add individual electric potentials to get total electric potential at the point
• Superposition works for both discrete point charges and continuous charge distributions
• Consider signs of charges and directions of electric fields from each charge distribution
  • Example: Dipole near a point charge, potential is sum of dipole potential and point charge potential

Electric Potential and Field Relationships

• Electric field is the negative gradient of electric potential
• The relationship between electric potential and electric field is path-independent
• Electric field lines always point from higher to lower potential regions