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8.4-1 Consider a magnetic field probe consisting of a flat circular loop of wire with radius 10 cm. The probe’s terminals correspond to a small gap in the loop. This probe is placed in a uniform magnetic field having magnitude B(t) = B0 sin(2πf t + α), where f = 100 kHz and α is an unknown constant. (By uniform, we mean that the magnetic field has the same magnitude and direction at all points in space.) The orientation of the loop with respect to the magnetic field vector is unknown. The voltage at the terminals is measured for all possible orientations of the probe, and it is found that the maximum voltage is 20 mV peakto-peak. What is B0?

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mavila18

Answer:

B_o = 1.013μT

Explanation:

To find B_o you take into account the formula for the emf:

[tex]\epsilon=-\frac{d\Phi_b}{dt}=-\frac{dBAcos\theta}{dt}=-Acos\theta\frac{dB}{dt}[/tex]

where you used that A (area of the loop) is constant, an also the angle between the direction of B and the normal to A.

By applying the derivative you obtain:

[tex]\epsilon=-Acos\theta (2\pi f) B_ocos(2\pi f t+ \alpha)[/tex]

when the emf is maximum the angle between B and the normal to A is zero, that is, cosθ = 1 or -1. Furthermore the cos function is 1 or -1. Hence:

[tex]\epsilon=2\pi fAB_o=2\pi (100*10^3Hz)(\pi (0.1m)^2)B_o=19739.20Hzm^2B_o\\\\B_o=\frac{20*10^{-3}V}{19739.20Hzm^2}=1.013*10^{-6}T=1.013\mu T[/tex]

hence, B_o = 1.013μT