A 1000-turn loop (radius = 0.032 m of wire is connected to a 15.0 ohm resistor a

Question

A 1000-turn loop (radius = 0.032 m of wire is connected to a 15.0 ohm resistor as shown in the figure. A magnetic field is directed perpendicular to the plane of the loop. The field points into the paper and has a magnitude that varies in time as B=gt, where g = 0.25 Tesla per second. Neglect the resistance of the wire. What is the magnitude of the potential difference between points a and b? 8.042 times 10^-1 V. What is the electrical energy dissipated in the resistor in 31.0 s? .47616 J Is the EMF induced in the coil constant? What is the relation between power and energy?

Explanation / Answer

Induced emf in a coil of N turns is given by = - N d/dt = - N d(BACos ) / dt where B is the magnetic field and A is the area of the loop and , the angle between the area vector A and the magnetic field B. In our case = 0

So, = - N d/dt = - N d(BACos ) / dt = - 1000 A dB/dt = - 1000 x x 0.0322 x d gt /dt = - 1000 x x 0.0322 x g

= - 1000 x x 0.0322 x 0.25 = 8.042477193 x 10-1  V

the negative sign just indicates that the induced emf tries to oppose the change in the magnetic flux.

As we see the induced emf is constant and has the same magnitude at all times since the magnetic field B is continuously changing as B = gt

Also, the power dissipated in a resistor is given as P = V2/R = 0.80424771932 / 15 = 0.0431209596 watt

Energy dissipated = power x time = 0.0431209596 x 31 = 1.336749748 J

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