The area of a rectangular loop of wire is $3.6 \times 10 ^ { - 3 } \mathrm {~m} ^ { 2 }$. The loop is placed in a uniform magnetic field that changes steadily from $0.20 \mathrm {~T}$ to $1.4 \mathrm {~T}$ in $1.6 \mathrm {~s}$. The plane of the loop is perpendicular to the direction of the magnetic field. What is the magnitude of the induced emf in that loop?
A) $3.0 \times 10 ^ { - 3 } \mathrm {~V}$
B) $0 \mathrm {~V}$
C) $1.8 \times 10 ^ { - 3 } \mathrm {~V}$
D) $2.8 \times 10 ^ { - 3 } \mathrm {~V}$
E) $2.7 \times 10 ^ { - 3 } \mathrm {~V}$

Question

Quizplus · Accepted Answer

The magnitude of the induced emf ($ \varepsilon $) in the loop can be calculated using Faraday's law of electromagnetic induction, which states that the induced emf is equal to the negative rate of change of the magnetic flux through the loop. Since the plane of the loop is perpendicular to the magnetic field, the magnetic flux ($ \Phi $) is given by $ \Phi = B \cdot A $, where $ B $ is the magnetic field strength and $ A $ is the area of the loop. The change in magnetic flux ($ \Delta \Phi $) is then $ \Delta B \cdot A $, where $ \Delta B = 1.4 \, \text{T} - 0.2 \, \text{T} = 1.2 \, \text{T} $. The rate of change of flux ($ \Delta \Phi / \Delta t $) is $ (1.2 \, \text{T} \cdot 3.6 \times 10^{-3} \, \text{m}^2) / 1.6 \, \text{s} $. Calculating this gives $ 2.7 \times 10^{-3} \, \text{V} $, hence the magnitude of the induced emf is $ 2.7 \times 10^{-3} \, \text{V} $.

The Area of a Rectangular Loop of Wire Is 3.6×10−3 m23.6 \times 10 ^ { - 3 } \mathrm {~m} ^ { 2 }3.6×10−3 m2

The Area of a Rectangular Loop of Wire Is $3.6 \times 10 ^ { - 3 } \mathrm {~m} ^ { 2 }$