Given data:
- Area of the coil (A) = 500 cm^2 = 0.05 m^2
- Number of turns in the coil (N) = 1000
- Magnetic field (B) = 0.4 G = 0.4 × 10^(-4) T
- Angle turned by the coil (θ) = π rad
- Time taken (t) = 1/10 sec

Calculating the magnetic flux:
The magnetic flux (Φ) through a coil is given by the formula:
Φ = B × A × cos(θ)

Given that the coil is held perpendicular to the magnetic field, the angle between the magnetic field and the area vector is 90 degrees. Therefore, cos(θ) = cos(90) = 0.

So, Φ = B × A × 0 = 0

Calculating the change in magnetic flux:
The change in magnetic flux (ΔΦ) is given by the formula:
ΔΦ = N × ΔB × A

Where ΔB is the change in magnetic field, which is 0 - B = -B.

So, ΔΦ = N × (-B) × A

Calculating the average induced e.m.f:
The average induced electromotive force (e.m.f) is given by the formula:
ε = -N × ΔΦ / Δt

Where Δt is the change in time, which is 1/10 sec.

Substituting the values:
ε = -N × (-B) × A / Δt
= N × B × A / Δt

Final Result:
The average induced e.m.f is N × B × A / Δt.

Explanation:
When a coil is placed in a magnetic field and is rotated, the change in magnetic flux through the coil induces an electromotive force (e.m.f) according to Faraday's law of electromagnetic induction. The magnitude of the induced e.m.f depends on the rate of change of the magnetic flux.

In this problem, the coil is held perpendicular to the magnetic field, resulting in no magnetic flux initially. As the coil is rotated, the magnetic flux through the coil increases from 0 to a maximum value. The change in magnetic flux induces an e.m.f in the coil.

To calculate the average induced e.m.f, we first determine the change in magnetic flux (ΔΦ) by considering the initial and final magnetic fields. Then, using the formula for average induced e.m.f, we obtain the final result.

Note: The negative sign in the formula for the average induced e.m.f indicates that the direction of the induced current will oppose the change in magnetic flux.