An Exponentially Decaying Magnetic Field Exponential decrease in the magnitude of the magnetic field through a loop with time. The induced emf and induced current in a conducting path attached to the loop vary with time in the same way. B Bax A loop of wire enclosing an area A is placed in a region where the magnetic field is perpendicular to the plane of the loop. The magnitude of B varies in time according to the expression 8 = 8, eat, where a is some constant. That is, at t=0, the field is B max and for t> 0, the field decreases exponentially (see the figure). Find the induced emf in the loop as a function of time. SOLUTION Conceptualize The physical situation is similar to that in the example Inducing an emf in a Coil except for two things: there is only one loop, and the field varies exponentially with time rather than linearly. Categorize We will evaluate the emf using Faraday's law from this section, so we categorize this example as a substitution ✓ problem. (Use the following as necessary: A. Bmax, and t.) Evaluate Faraday's law for the situation described here: dB-- AB max =-AB max dt This expression indicates that the induced emf decays exponentially in time. The maximum emf occurs at t=0, where EXERCISE The plot of Ɛ versus t is similar to the B-versus-t curve shown in the figure. A flat loop of wire consisting of a single turn of cross-sectional area 7.10 cm² is perpendicular to a magnetic field that increases uniformly in magnitude from 1.95 T to 4.80 T in 1.65 s. What is the magnitude of the resulting induced current (in mA) if the loop has a resistance of 4.90 ? Hint MA
An Exponentially Decaying Magnetic Field Exponential decrease in the magnitude of the magnetic field through a loop with time. The induced emf and induced current in a conducting path attached to the loop vary with time in the same way. B Bax A loop of wire enclosing an area A is placed in a region where the magnetic field is perpendicular to the plane of the loop. The magnitude of B varies in time according to the expression 8 = 8, eat, where a is some constant. That is, at t=0, the field is B max and for t> 0, the field decreases exponentially (see the figure). Find the induced emf in the loop as a function of time. SOLUTION Conceptualize The physical situation is similar to that in the example Inducing an emf in a Coil except for two things: there is only one loop, and the field varies exponentially with time rather than linearly. Categorize We will evaluate the emf using Faraday's law from this section, so we categorize this example as a substitution ✓ problem. (Use the following as necessary: A. Bmax, and t.) Evaluate Faraday's law for the situation described here: dB-- AB max =-AB max dt This expression indicates that the induced emf decays exponentially in time. The maximum emf occurs at t=0, where EXERCISE The plot of Ɛ versus t is similar to the B-versus-t curve shown in the figure. A flat loop of wire consisting of a single turn of cross-sectional area 7.10 cm² is perpendicular to a magnetic field that increases uniformly in magnitude from 1.95 T to 4.80 T in 1.65 s. What is the magnitude of the resulting induced current (in mA) if the loop has a resistance of 4.90 ? Hint MA
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Transcribed Image Text:An Exponentially Decaying Magnetic Field
Exponential decrease in the magnitude of the
magnetic field through a loop with time. The
induced emf and induced current in a conducting
path attached to the loop vary with time in the
same way.
B
Bymax
A loop of wire enclosing an area A is placed in a region where the magnetic field is perpendicular to the plane of the loop. The magnitude of B varies in time according to the expression B = B
exponentially (see the figure). Find the induced emf in the loop as a function of time.
e-at
", where a is some constant. That is, at t = 0, the field is B
and for t > 0, the field decreases
max
SOLUTION
Conceptualize The physical situation is similar to that in the example Inducing an emf in a Coil except for two things: there is only one loop, and the field varies exponentially with time rather than linearly.
Categorize We will evaluate the emf using Faraday's law from this section, so we categorize this example as a substitution
problem.
(Use the following as necessary: A, B max a, and t.)
Evaluate Faraday's law for the situation described here:
J&B = -
8 = -
dt
8 =
d
dt
d-at
(AB max=-AB max t
This expression indicates that the induced emf decays exponentially in time. The maximum emf occurs at t = 0, where &
EXERCISE
. The plot of Є versus t is similar to the B-versus-t curve shown in the figure.
max
A flat loop of wire consisting of a single turn of cross-sectional area 7.10 cm is perpendicular to a magnetic field that increases uniformly in magnitude from 1.95 T to 4.80 T in 1.65 s. What is the magnitude of the resulting induced current (in mA) if the loop has a resistance of 4.90 ?
Hint
MA
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