In Figure (a), a uniform magnetic field increases in magnitude with time t as given by Figure (b), where the vertical axis scale is set by and the horizontal scale is set by A circular conducting loop of area lies in the field, in the plane of the page. The amount of charge q passing point A on the loop is given in Figure (c) as a function of t, with the vertical axis scale set by and the horizontal axis scale again set by localid="1661854094654" . What is the loop’s resistance?
Resistance of the loop is, R =
Calculate the induced emf by using the slope of the line form Fig. 30-42(b) in Faraday’s law. Find the current by finding the slope of the line from Fig.30-42 (c) to find the resistance of the loop by using Ohm’s law.
Faraday's law of electromagnetic induction states, Whenever a conductor is placed in a varying magnetic field, an electromotive force is induced in it.
Ohm's law states that the voltage across a conductor is directly proportional to the current flowing through it, provided all physical conditions and temperatures remain constant.
Formulae are as follow:
Where, is magnetic flux, dt is time, dq is charge, i is current, R is resistance, 𝜀 is emf,
First, find the emf by using Faraday’s law.
From Fig.30 - 42 (b) the slope of the line i.e. dB/dt can be found as,
Faraday’s law is given as,
As magnetic field is perpendicular to the area vector,
Substituting the values,
Now, the current can be calculated from Fig. 30- 42 (c).
The slope of the graph gives dq/dt.
Now, relate the induced emf to resistance and current by using Ohm’s law.
Substituting the values,
Hence, resistance of the loop is,
Therefore, we calculated the resistance of the loop by using the slopes from the graph, Faraday’s law, and Ohm’s law.
In Fig. 30-26, a wire loop has been bent so that it has three segments: segment bc (a quarter-circle), ac (a square corner), and ab (straight). Here are three choices for a magnetic field through the loop:
where B is in milliteslas and t is in seconds. Without written calculation, rank the choices according to (a) the work done per unit charge in setting up the induced current and (b) that induced current, greatest first. (c) For each choice, what is the direction of the induced current in the figure?
In Figure, . Immediately after switch S is closed, (a) what is i1? (b) what is i2? (Let currents in the indicated directions have positive values and currents in the opposite directions have negative values.) A long time later, (c) what is i1? (d) what is i2? The switch is then reopened. Just then, (e) what is i1? (f) what is i2? A long time later, (g) what is i1? (h) what is i2?
In Figure, a wire forms a closed circular loop, of radius R = 2.0m and resistance . The circle is centered on a long straight wire; at time t = 0, the current in the long straight wire is 5.0 A rightward. Thereafter, the current changes according to . (The straight wire is insulated; so there is no electrical contact between it and the wire of the loop.) What is the magnitude of the current induced in the loop at times t > 0?
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