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Magnetic flux ϕ linked with a stationary loop of resistance R varies with time t as ϕ = a t ( T – t ) . The amount of heat generated in the loop, during the time interval T , is

Options

  1. Aa T 3 R
  2. Ba 2 T 2 3 R
  3. Ca 2 T 2 R
  4. Da 2 T 3 3 R

Correct answer

D. a 2 T 3 3 R

Step-by-step solution

| E | = d ϕ d t = ( a T – 2 a t ) H = ∫ E 2 R d t = a 2 R ∫ 0 T T - 2 t 2 d t = a 2 R ∫ 0 T T 2 + 4 t 2 - 4 T t d t = a 2 R T 2 t 0 T + 4 3 t 3 0 T - 2 T t 2 0 T = a 2 T 3 3 R

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