Definite Integration
Average value of a function
Grade 12
Question:
<p>The average value of the electromotive force \(E_m\) over the duration \(t \in [0, T]\) where the electromotive force \(E\) is computed by \(E = E_0 \sin\frac{2\pi t}{T}\), \(E_0\) and \(T\) being constants, is equal to</p>
<p>(a) \(E_0\)</p>
<p>(b) 0</p>
<p>(c) \(\dfrac{E_0}{2\pi}\)</p>
<p>(d) none of these</p>
Step-by-Step Solution
Key Concept: The average value of a function f(t) over [a,b] is defined as (1/(b-a))∫[a to b]f(t)dt. Here we apply this formula with f(t) = E₀sin(2πt/T) over [0,T].
<p><strong>Step 1:</strong> Recall the average value formula: E_avg = (1/(T-0))∫₀ᵀ E dt = (1/T)∫₀ᵀ E₀sin(2πt/T) dt</p><p><strong>Step 2:</strong> Factor out constant E₀: E_avg = (E₀/T)∫₀ᵀ sin(2πt/T) dt</p><p><strong>Step 3:</strong> Substitute u = 2πt/T, so du = (2π/T)dt, giving dt = (T/2π)du. When t=0, u=0; when t=T, u=2π</p><p><strong>Step 4:</strong> E_avg = (E₀/T) · (T/2π)∫₀²π sin(u) du = (E₀/2π)[−cos(u)]₀²π</p><p><strong>Step 5:</strong> E_avg = (E₀/2π)[−cos(2π) + cos(0)] = (E₀/2π)[−1 + 1] = 0</p><p><strong>Physical insight:</strong> Over one complete period, the sinusoidal EMF spends equal time positive and negative, so the average value is zero.</p><p>∴ Answer: B (which is 0)</p>
Correct Answer: B