University:
Santa Fe College
Course:
ETS 2527C | Electromechanical Components and Mechanisms
Academic year:

2024
Views:

291

Pages:

3
Author:

Jabin F.

# Electric potential (V) = (Work done by charge) / q V = W / q E⃗ = F⃗ / q V_A - V_B = -∫_B^A E⃗ · d⃗l V(r) - V(∞) = V(r) = ∫_∞^r E⃗ · d⃗r Infinite can not take as reference point if charge it self is extending to infinite. e.g. Infinitely long charge wire. e.g. Infinitely large plane sheet. Potential through charge (when Gauss' law is applicable) V = q / (4πε₀r) = (1 / 4πε₀) ∫_C λ dl / r V = (1 / 4πε₀) ∫_S σ ds / r = (1 / 4πε₀) ∫_C λ dl / r Ques. 26 I-method for (r ≥ a) V(r) = -∫_∞^r E⃗ · d⃗l d⃗l = dr r̂ + r dθ θ̂ V(r) = -∫_∞^r E(r) r̂ · (dr r̂ + r dθ θ̂) = -∫_∞^r E(r) dr V(r) = -∫_∞^b E(r) dr - ∫_b^r E(r) dr V(r) = -∫_∞^b (K(b-a) / ε₀r²) dr - ∫_b^r (K(r-a) / ε₀r²) dr V(r) = (K(b-a) / ε₀) [ (1/r) ]_∞^b - (K / ε₀) [ ln(r) + (a/r) ]_b^a = (K(b-a) / ε₀) (1/b) - (K / ε₀) [ ln(a) + (a/a) - ln(b) - (a/b) ] V(r) = (K / ε₀) [ ln(b/a) - (a/b) + (1/b) ] # a < r < b V = -∫_∞^r E dr = -∫_∞^b (K(b-a) / ε₀r²) dr - ∫_b^r (K(r-a) / ε₀r²) dr V = -∫_∞^b (K(b-a) / ε₀r²) dr - ∫_b^r (K(r-a) / ε₀r²) dr = (K(b-a) / ε₀) [ (1/r) ]_∞^b - (K / ε₀) [ ln(r) + (a/r) ]_b^a = (K(b-a) / ε₀b) - (K / ε₀) [ ln(a) + 1 - ln(b) - (a/b) ] = (K / ε₀) [ (b-a)/b - ln(a) - 1 + ln(b) + (a/b) ] = (K / ε₀) [ 1 - (a/b) + ln(b/a) ] # r > b V = -∫_∞^r E dr = -∫_∞^b (K(b-a) / ε₀r²) dr = (K(b-a) / ε₀) [ (1/r) ]_∞^b = (K(b-a) / ε₀b)