A flat, rectangular coil of dimensions l and w is pulled with uni-form speed v through a uniform magnetic field B with the plane of its area perpen-dicular 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. Identify the area of the coil, which can be calculated using the formula for the area of a rectangle, A = l \times w, where l is the length and w is the width of the coil.
2. Understand that the magnetic flux (\Phi) through the coil is given by \Phi = B \times A, where B is the magnetic field strength and A is the area of the coil. Since the coil's plane is perpendicular to the magnetic field, the angle \theta between the magnetic field and the normal to the area of the coil is 0 degrees, making the cosine of \theta equal to 1.
3. Calculate the rate of change of magnetic flux to find the induced electromotive force (emf). The emf (\mathcal{E}) induced in the coil can be calculated using Faraday's Law of Electromagnetic Induction, which states \mathcal{E} = -\frac{d\Phi}{dt}. Since the coil is moving at a uniform speed v, and the width w is perpendicular to the velocity, the change in flux over time can be expressed as \frac{d\Phi}{dt} = B \times w \times v.
4. For part (b), if the speed v and the magnetic field B are both tripled, substitute 3v for v and 3B for B in the expression for \frac{d\Phi}{dt} from step 3. This results in \frac{d\Phi}{dt} = (3B) \times w \times (3v) = 9Bwv.
5. Recognize that the induced emf is proportional to the rate of change of magnetic flux. Therefore, tripling both the speed and the magnetic field results in the induced emf being multiplied by nine times the original emf calculated in part (a).

Key Concepts

Faraday's Law of Electromagnetic Induction

Faraday's Law states that the electromotive force (emf) induced in a closed loop is directly proportional to the rate of change of magnetic flux through the loop. In this scenario, as the coil moves through the magnetic field, the area exposed to the field changes, leading to a change in magnetic flux and thus inducing an emf.

Magnetic Flux

Magnetic flux is defined as the product of the magnetic field strength (B) and the area (A) through which the field lines pass, taking into account the angle between the field and the normal to the surface. In this case, since the coil's plane is perpendicular to the magnetic field, the magnetic flux is maximized, which is crucial for calculating the induced emf.

Induced EMF and Speed Relationship

The induced emf in a coil moving through a magnetic field is influenced by the speed of the coil. According to Faraday's Law, if the speed (v) of the coil is increased, the rate at which the magnetic flux changes also increases, leading to a higher induced emf. Therefore, tripling both the speed and the magnetic field will result in a tripling of the induced emf.