How fast must a 1000 -turn coil (each with a 20 cm^2 area) turn in the Earth's magnetic field o

step 1 Hi, for this question, want to find how fast the 1 ,000 turn coil. So N will be 1 ,000 turns. Ar

How fast must a 1000 -turn coil (each with a 20 cm^2 area)...

How fast must a 1000 -turn coil (each with a $20 \mathrm{~cm}^{2}$ area) turn in the Earth's magnetic field of $0.70 \mathrm{G}$ to generate a voltage that has a maximum value (i.e., an amplitude) of $0.50 \mathrm{~V}$ ? We assume the coil's axis to be oriented in the field so as to give maximum flux change when rotated. Then $B=7.0 \times 10^{-5} \mathrm{~T}$ in the expression $$ \mathscr{E}=2 \pi N A B f \cos 2 \pi f t $$ Because $\cos 2 \pi f t$ has a maximum value of unity, the amplitude of the voltage is 2\piNBAf. Therefore, $$ f=\frac{0.50 \mathrm{~V}}{2 \pi N A B}=\frac{0.50 \mathrm{~V}}{(2 \pi)(1000)\left(20 \times 10^{-4} \mathrm{~m}^{2}\right)\left(7.0 \times 10^{-5} \mathrm{~T}\right)}=0.57 \mathrm{kHz} $$

How fast must a 1000 -turn coil (each with a $20 \mathrm{~cm}^{2}$ area) turn in the Earth's magnetic field of $0.70 \mathrm{G}$ to generate a voltage that has a maximum value (i.e., an amplitude) of $0.50 \mathrm{~V}$ ?

We assume the coil's axis to be oriented in the field so as to give maximum flux change when rotated. Then $B=7.0 \times 10^{-5} \mathrm{~T}$ in the expression
$$
\mathscr{E}=2 \pi N A B f \cos 2 \pi f t
$$
Because $\cos 2 \pi f t$ has a maximum value of unity, the amplitude of the voltage is 2\piNBAf. Therefore,
$$
f=\frac{0.50 \mathrm{~V}}{2 \pi N A B}=\frac{0.50 \mathrm{~V}}{(2 \pi)(1000)\left(20 \times 10^{-4} \mathrm{~m}^{2}\right)\left(7.0 \times 10^{-5} \mathrm{~T}\right)}=0.57 \mathrm{kHz}
$$

Key Concepts

Angular Frequency and Frequency

In oscillatory systems, the angular frequency (?) relates to the standard frequency (f) through the equation ? = 2?f. This relation is crucial when converting the time derivatives in sinusoidal functions into expressions that involve the frequency of rotation, helping to determine the peak induced voltage.

Rotational Motion in a Magnetic Field

When a coil rotates within a magnetic field, the component of the magnetic field that threads the coil changes with time, producing a time-varying flux. This variation leads to a sinusoidally varying induced voltage, and understanding this motion is key to relating the rotation speed to the voltage amplitude.

Unit Conversion in Electromagnetism

Accurate problem solving in electromagnetism often requires converting units into the standard SI system. For example, converting magnetic field strength from gauss to tesla ensures that all terms in the formulas are consistent and the calculations yield correct results.

Faraday's Law of Electromagnetic Induction

This principle states that a change in magnetic flux through a circuit induces an electromotive force (EMF) in that circuit. It forms the basis for understanding how variations in magnetic environments (such as a rotating coil in a magnetic field) can produce electricity.

Magnetic Flux

Magnetic flux quantifies the amount of magnetic field passing through a given area and is calculated as the product of the magnetic field strength, the area of the coil, and the cosine of the angle between the magnetic field and the normal to the area. It is the change in this flux that drives the induced EMF.

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