A truck drives onto a loop detector and increases the...

A truck drives onto a loop detector and increases the downward-pointing component of the magnetic field within the loop from $1.2 \times 10^{-5}$ T to a larger value, $B$, in $0.38 \mathrm{~s}$. The detector is circular, has a radius of $0.67 \mathrm{~m}$, and consists of three loops of wire. What is $B$, given that the induced emf is $8.1 \times 10^{-4} \mathrm{~V} ?$ A. $3.6 \times 10^{-5} \mathrm{~T}$ B. $7.3 \times 10^{-5} \mathrm{~T}$ C. $8.5 \times 10^{-5} \mathrm{~T}$ D. $24 \times 10^{-5} \mathrm{~T}$

A truck drives onto a loop detector and increases the downward-pointing component of the magnetic field within the loop from $1.2 \times 10^{-5}$ T to a larger value, $B$, in $0.38 \mathrm{~s}$. The detector is circular, has a radius of $0.67 \mathrm{~m}$, and consists of three loops of wire. What is $B$, given that the induced emf is $8.1 \times 10^{-4} \mathrm{~V} ?$
A. $3.6 \times 10^{-5} \mathrm{~T}$ B. $7.3 \times 10^{-5} \mathrm{~T}$
C. $8.5 \times 10^{-5} \mathrm{~T}$
D. $24 \times 10^{-5} \mathrm{~T}$

Key Concepts

Magnetic Flux

Magnetic flux is a measure of the total magnetic field passing perpendicularly through a given area. It is calculated as the product of the magnetic field strength and the area through which it penetrates, taking into account the angle between the field and the normal to the surface. Magnetic flux is central to analyzing problems involving loops and coils in magnetic fields.

Faraday's Law of Electromagnetic Induction

Faraday's Law states that any change in the magnetic flux through a circuit induces an electromotive force (emf) in the circuit. The induced emf is directly proportional to the rate at which the magnetic flux changes and the number of loops in the circuit, with the negative sign reflecting Lenz's Law. This principle underpins the operation of many electrical devices and explains how energy can be generated via magnetic fields.

Induced emf in Multi-Turn Loops

When a circuit consists of multiple turns or loops, the total induced emf is the sum of the contributions from each individual loop. This means that the effective induced voltage is multiplied by the number of turns. Recognizing this helps in calculations where induced emf is measured over several loops, emphasizing the dependence on both the number of turns and the change in magnetic flux.

Rate of Change of Magnetic Field

The rate of change of the magnetic field is key to determining how quickly the magnetic flux through an area changes over time. This metric is crucial when applying Faraday's Law because the induced emf is proportionate to the time derivative of the magnetic flux. In problems involving moving magnetic sources or time-varying fields, it identifies how dynamic changes in the field contribute to electrical phenomena.

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