Lecture Notes: Electric Field and Electric Potential

Introduction

• Charges create electric fields and electric potentials.
• Direction of electric fields:
  • Radially outward for positive charges.
  • Radially inward for negative charges.
• Electric potential (V):
  • Created everywhere around a charge.
  • Is a scalar, just a number, not a vector.

Electric Field and Electric Potential Formulas

• Formulas involve plugging in distance (R) and the charge (Q) to find electric fields and potentials.
• For electric potential, you don't square R, and vector nature isn't a concern.

Distributed Charges

• Uniform Charge Distribution:
  • Charge distributed over a line or area requires integration to find E and V.
  • Use formula for point charge but break into infinitesimal charges (dQ).
• Integration Approach:
  • Define differential charge (dQ) and calculate its contribution to the electric field (dE).
  • Add up contributions from all differential elements using integration.

Line Charge Example

• Electric Field Calculation:
  • Charge spread over a line, find E at a point X from the rod.
  • Use integration due to distributed nature.
  • Define charge per length (λ = Q/L).
  • Integrate dQ = λdL to find total electric field.
  • Integral limits depend on geometry of the charge distribution.

Curved Charge Example (Semicircle)

• Electric Potential Calculation:
  • Distance from all points on a semicircle to center is constant.
  • Simplified: V = Q / (4πε₀R).
• Electric Field Calculation:
  • Use integration considering vector nature.
  • Find horizontal component (Ex) due to symmetry.
  • Integrate dq * cos(θ) to find Ex.
• Variables and Limits:
  • Convert dq to λdl, where dl = Rdθ (arc length).
  • Integration limits based on angular position (θ).
  • Simplified due to symmetry and constant radius.*

Key Concepts

• Integration:
  • Necessary for non-point charge distributions.
  • Adjust variables and limits based on geometry.
• Vector and Scalar Quantities:
  • Electric field (E) is a vector; direction matters.
  • Electric potential (V) is a scalar; direction does not matter.

Recap

• Use dq instead of q and integrate to find E for distributed charges.
• Adjust dq to match the geometry and given problem data.
• For V, use scalar simplification when applicable.