A magnetic field passes through a stationary wire loop, and itsmagnitude changes

Question

A magnetic field passes through a stationary wire loop, and itsmagnitude changes in time according to the graph in the drawing.The direction of the field remains constant, however. There arethree equal time intervals indicated in the graph: 0– 3.0 s, 3.0 – 6.0 s,and 6.0 – 9.0 s. The loop consists of 42turns of wire and has an area of 0.15 m2. The magneticfield is oriented parallel to the normal to the loop. For purposesof this problem, this means that =0° in Equation22.2. (a) For each interval, determinethe induced emf. (b) The wire has aresistance of 0.50 . Determine the inducedcurrent for the first and third intervals.

Explanation / Answer

   total area of loop = 42 x 0.15 = 6.3 m2    e.m.f induced = - d/dt    = area x B x cos = area x B = 6.3B    d/dt = 6.3 dB/dt     e.m.f., = - 6.3 dB/dt Now e.m.f induced in the three cases depends on how B varieswith time, t a) for time 0 - 3 sec       here variation of B is linear with t    its a straight line curve    the equation of straight line is:         (B- 0) = (0.4 - 0)/(3 -0) x (t - 0)         B = 0.4t/3        dB/dt = 0.4/3    = - 6.3 dB/dt = -6.3 x 0.4 / 3 = - 0.84 V       for 3 - 6 sec      here B is constant with time      so dB/dt = 0        = - 6.3 dB/dt =0 V    for time 6 - 9 sec       here variation of B is linear with t    its a straight line curve    the equation of straight line is:         (B- 0.2) = (0.4 -0.2)/(6 - 9) x (t - 9)         B = 0.2[1 - (t -9)/3]        dB/dt = - 0.2/3    = - 6.3 dB/dt = -6.3 x -0.2 / 3 = 0.42 V   b) I1 = 0.84/0.5 = 1.68 A      I2 = 0.42/0.5 = 0.84 A

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