The induced EMF in a coil with N turns is given by which expression?

• A. e = N dΦ/dt
• B. e = - dΦ/dt
• C. e = Φ dN/dt
• D. e = - N dΦ/dt

Answer: (D)

When a coil has N turns, the magnetic flux linked by the coil is the flux through a single turn times the number of turns, λ = NΦ. Faraday’s law says the induced emf is the negative time rate of change of this flux linkage: e = - dλ/dt = - d(NΦ)/dt. If the number of turns is constant, this becomes e = - N dΦ/dt. The minus sign reflects Lenz’s law, meaning the induced emf opposes the change in flux. If the turns could change with time, the full expression would be e = - (N dΦ/dt + Φ dN/dt), but for a fixed coil the standard result is e = - N dΦ/dt.