Within the green dashed circle shown in the figure below, the magnetic field cha

Question

Within the green dashed circle shown in the figure below, the magnetic field changes with time according to the expression B = 10.00t^3 - 2.00t^2 + 0.800, where B is in teslas, t is in seconds, and R = 2.25 cm. When t = 2.00s, calculate the magnitude of the force exerted on an electron located at point P1 , which is at a distance r1 = 4.50 cm from the center of the circular field region. N When t = 2.00s, calculate the direction of the force exerted on an electron located at point p2, which is at a distance r2 = 4.50 cm from the center of the circular field region. At what instant is this force equal to zero?(Consider the time after t = 0 s.)

Explanation / Answer

Emf = integral( E*ds) = - d?B/dt

d?B/dt = B*A

A = (pi*R^2)

So, we can now solve for E, by taking the derivative of Emf

integral( E*ds) =B*A

E = dB*A/ds

**note ds = 2pi*r1

We find dB by taking the derivative of B:

B=10.00t3 - 2.00t2 + 0.800,

dB = 30.00t2- 2.00t, at time t=2.0 sec, so

dB = 30*(2.0)2-2*(2.0)

dB = 116

Plugging dB and A into the equation for E we get:

E = 116*(pi*R^2) /(2pi*r1)

R = 2.25 cm = 2.25*10^-2 m

r1 = 4.5 cm = 4.5*10^-2 m

so,

E = 116*(pi*(2.25*10^-2 m)^2) /(2pi*4.5*10^-2 m)

Force on the electron   F   = eE  

here   e = charge of the electron = 1.6*10^-19 C

Force = (116*(pi*(2.25*10^-2 m)^2) /(2pi*4.5*10^-2 m) )*(1.6*10^-19 C)

F = 1.044x10^-19 N

Using Right Hand Rule, we know this force is clockwise

Part (c)

The force is equal to zero dB/dt = 0

dB = 30.0t2- 2.0t = 0

t = 1/15 = 0.0667 s

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