A magnetic field B exists inside a circular region of radius a whose normal is along the z-axis as shown in Fig. 3. The direction of B makes an angle 0 with the z-axis. The value of B varies with timet according to B(r, t) = Bor?cos(wt) where r is the radial coordinate from the center of the circle, w is the constant angular frequency and B, a constant amplitude. %3D a FIGURE 3 (a) Find the magnetic flux through the circle. Using your result, determine the magnetic flux through the circle at time t = 0. %3D (b) Determine the value of the induced EMF at wt = 37/2. (c) If the rim of the circle is a metal loop having resistance R, what is the induced current I in the loop? Using Lenz's law, determine the flow direction of the induced current in this wire at times wt = 0 and wt = T/2. (d) Following (c), the induced current I in the metal loop will generate its own magnetic field B;. In class we derived a formula for the magnetic field at the central axis of a circular current by Biot- Savart law. Apply that formula to write down B; along the central axis of the circular loop as a function of time t. Hint: You may find the derivation for B; referenced above in lecture 21 (as well as Chapter 12 of the textbook). (N
A magnetic field B exists inside a circular region of radius a whose normal is along the z-axis as shown in Fig. 3. The direction of B makes an angle 0 with the z-axis. The value of B varies with timet according to B(r, t) = Bor?cos(wt) where r is the radial coordinate from the center of the circle, w is the constant angular frequency and B, a constant amplitude. %3D a FIGURE 3 (a) Find the magnetic flux through the circle. Using your result, determine the magnetic flux through the circle at time t = 0. %3D (b) Determine the value of the induced EMF at wt = 37/2. (c) If the rim of the circle is a metal loop having resistance R, what is the induced current I in the loop? Using Lenz's law, determine the flow direction of the induced current in this wire at times wt = 0 and wt = T/2. (d) Following (c), the induced current I in the metal loop will generate its own magnetic field B;. In class we derived a formula for the magnetic field at the central axis of a circular current by Biot- Savart law. Apply that formula to write down B; along the central axis of the circular loop as a function of time t. Hint: You may find the derivation for B; referenced above in lecture 21 (as well as Chapter 12 of the textbook). (N
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Transcribed Image Text:A magnetic field B exists inside a circular region of radius a whose normal is along the z-axis as shown in
Fig. 3. The direction of B makes an angle 0 with the z-axis. The value of B varies with time t according
to B(r,t) = B,r²cos(wt) where r is the radial coordinate from the center of the circle, w is the constant
angular frequency and B, a constant amplitude.
В
a
FIGURE 3
(a) Find the magnetic flux through the circle. Using your result, determine the magnetic flux through
the circle at time t = 0.
(b) Determine the value of the induced EMF at wt = 31/2.
(c) If the rim of the circle is a metal loop having resistance R, what is the induced current I in the
loop? Using Lenz's law, determine the flow direction of the induced current in this wire at times
wt =
O and wt
= T/2.
(d) Following (c), the induced current I in the metal loop will generate its own magnetic field B;.
class we derived a formula for the magnetic field at the central axis of a circular current by Biot-
Savart law. Apply that formula to write down B; along the central axis of the circular loop as a
In
function of time t.
Hint: You may find the derivation for B; referenced above in lecture 21 (as well as Chapter 12 of
the textbook).
(N
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