Figure 30-39 shows a closed loop of wire that consists of a...

Figure $30-39$ shows a closed loop of wire that consists of a pair of equal semicircles, of radius $3.7 \mathrm{cm},$ lying in mutually perpendicular planes. The loop was formed by folding a flat circular loop along a diameter until the two halves became perpendicular to each other. A uniform magnetic field $\vec{B}$ of magnitude 76 $\mathrm{mT}$ is directed perpendicular to the fold diameter and makes equal angles (of $45^{\circ}$ ) with the planes of the semicircles. The magnetic field is reduced to zero at a uniform rate during a time interval of 4.5 $\mathrm{ms}$ . During this interval, what are the (a) magnitude and (b) direction (clockwise or counterclockwise when viewed along the direction of $\vec{B}$ ) of the emf induced in the loop?

Figure $30-39$ shows a closed
loop of wire that consists of a pair of
equal semicircles, of radius $3.7 \mathrm{cm},$
lying in mutually perpendicular
planes. The loop was formed by folding a flat circular loop along a diameter until the two halves became
perpendicular to each other. A uniform magnetic field $\vec{B}$ of magnitude
76 $\mathrm{mT}$ is directed perpendicular to
the fold diameter and makes equal
angles (of $45^{\circ}$ ) with the planes of the
semicircles. The magnetic field is reduced to zero at a uniform rate
during a time interval of 4.5 $\mathrm{ms}$ . During this interval, what are
the (a) magnitude and (b) direction (clockwise or counterclockwise when viewed along the direction of $\vec{B}$ ) of the emf induced in
the loop?

Key Concepts

Geometry and Orientation Effects

The geometry and orientation of a conductive loop play a critical role in determining the effective area experiencing the magnetic flux. When parts of the loop lie in different planes or are folded, the projection of the loop area onto a plane perpendicular to the magnetic field must be considered. This geometric factor directly influences both the magnitude of the magnetic flux and the resulting induced emf as per Faraday's law.

Lenz's Law

Lenz’s law provides the direction of the induced current resulting from a change in magnetic flux. It asserts that the induced current will flow in such a way as to oppose the change in flux that produced it. This opposition is a manifestation of the conservation of energy and helps determine whether the induced current is clockwise or counterclockwise when viewed from a particular angle relative to the magnetic field.

Faraday's Law of Electromagnetic Induction

Faraday's law states that a changing magnetic flux through a circuit induces an electromotive force (emf) in that circuit. This induced emf is proportional to the rate of change of the magnetic flux and is given by the negative derivative of the flux with respect to time. The law is fundamental in explaining how variations in magnetic fields can generate currents in conductive loops.

Magnetic Flux

Magnetic flux is a measure of the amount of magnetic field passing through a given area. It is calculated as the dot product of the magnetic field vector and the area vector of the loop. The flux depends on both the magnitude of the magnetic field and the orientation of the loop relative to the field direction, meaning that any change in either the field strength or the area’s orientation will alter the magnetic flux through the loop.

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