A flat, rectangular coil of dimensions l and w is pulled with uniform speed v through a uniform magnetic field B with the plane of its area perpendicular to the field (Fig. E29.14). (a) Find the emf induced in this coil. (b) If the speed and magnetic field are both tripled, what is the induced emf?

Verified step by step guidance

1. Understand that the problem involves electromagnetic induction, where a change in magnetic flux through a coil induces an electromotive force (emf). The coil is moving through a magnetic field, which changes the flux.
2. Recall Faraday's Law of Induction, which states that the induced emf (\( \mathcal{E} \)) in a coil is equal to the negative rate of change of magnetic flux (\( \Phi_B \)) through the coil: \( \mathcal{E} = -\frac{d\Phi_B}{dt} \).
3. Calculate the magnetic flux (\( \Phi_B \)) through the coil. Since the magnetic field (\( B \)) is uniform and perpendicular to the plane of the coil, \( \Phi_B = B \cdot A \), where \( A = l \cdot w \) is the area of the coil.
4. Determine the rate of change of magnetic flux. As the coil moves with speed \( v \), the area exposed to the magnetic field changes. The rate of change of flux is \( \frac{d\Phi_B}{dt} = B \cdot w \cdot v \), since the length \( l \) of the coil is moving out of the field at speed \( v \).
5. For part (b), if both the speed \( v \) and the magnetic field \( B \) are tripled, substitute \( 3v \) and \( 3B \) into the expression for the rate of change of flux: \( \frac{d\Phi_B}{dt} = 3B \cdot w \cdot 3v \). Calculate the new induced emf using this updated rate of change.

Key Concepts

Faraday's Law of Electromagnetic Induction

Faraday's Law states that the induced electromotive force (emf) in a closed loop is equal to the negative rate of change of magnetic flux through the loop. For a coil moving through a magnetic field, the emf can be calculated using the formula emf = -dΦ/dt, where Φ is the magnetic flux.

Magnetic Flux

Magnetic flux (Φ) is the measure of the quantity of magnetism, considering the strength and extent of a magnetic field. It is calculated as Φ = B * A * cos(θ), where B is the magnetic field strength, A is the area through which the field lines pass, and θ is the angle between the field lines and the perpendicular to the surface.

Effect of Velocity and Magnetic Field on Induced EMF

The induced emf in a coil moving through a magnetic field is directly proportional to the velocity of the coil and the strength of the magnetic field. If both the speed of the coil and the magnetic field are tripled, the induced emf will increase by a factor of nine, as emf = B * l * v for a rectangular coil moving perpendicular to the field.