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Where is ∂B∂tnonzero in Figure 7.21(b)? Exploit the analogy between Faraday's law and Ampere's law to sketch (qualitatively) the electric field.

Short Answer

Expert verified

The term ∂B∂tis nonzero along the left and right edges of the shaded rectangle.

Step by step solution

01

Define Faraday's Law.

Faraday's law is the most fundamental law in Electromagnetic induction, which depicts that whenever a conductor is placed in the magnetic field, which is variable in nature, then an EMF will be induced in the conductor if the conductor has been short circuited, then current will flow in the conductor.

02

Determine the nonzero position of ∂B∂t

The figure concludes that the (inward) flux through the strip on the left isincreasing. On the contrary, the (inward) flux passing through the strip on the right decreases, similar to the two current sheets under Ampere's circuital law.

Now, withB→E andμ0Ienc→−»åÏ•dt from equation 7.19, the significance of the negative sign is that the current has been flown out to one on the left, so its field is counter clockwise, and the one on the right is like a current flowing in, so it field is clockwise.

The diagram of the above description is shown below:

Therefore, the term∂B∂t is nonzero along the left and right edges of the shaded rectangle.

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Most popular questions from this chapter

A square loop, side a , resistance R , lies a distance from an infinite straight wire that carries current l (Fig. 7.29). Now someone cuts the wire, so l drops to zero. In what direction does the induced current in the square loop flow, and what total charge passes a given point in the loop during the time this current flows? If you don't like the scissors model, turn the current down gradually:

I(t)={(1-t)I0for0≤t≤1/afort>/a

Two long, straight copper pipes, each of radius a, are held a distance 2d apart (see Fig. 7.50). One is at potential V0, the other at -V0. The space surrounding the pipes is filled with weakly conducting material of conductivity σ. Find the current per unit length that flows from one pipe to the other. [Hint: Refer to Prob. 3.12.]

In a perfect conductor, the conductivity is infinite, so E=0(Eq. 7.3), and any net charge resides on the surface (just as it does for an imperfect conductor, in electrostatics).

(a) Show that the magnetic field is constant (∂B∂t=0), inside a perfect conductor.

(b) Show that the magnetic flux through a perfectly conducting loop is constant.

A superconductor is a perfect conductor with the additional property that the (constant) B inside is in fact zero. (This "flux exclusion" is known as the Meissner effect.)

(c) Show that the current in a superconductor is confined to the surface.

(d) Superconductivity is lost above a certain critical temperature (Tc), which varies from one material to another. Suppose you had a sphere (radius ) above its critical temperature, and you held it in a uniform magnetic field B0z^while cooling it below Tc. Find the induced surface current density K, as a function of the polar angleθ.

A long solenoid, of radius a, is driven by an alternating current, so that the field inside is sinusoidal:Bt=B0cosÓ¬tz^. A circular loop of wire, of radius a/2 and resistance R , is placed inside the solenoid, and coaxial with it. Find the current induced in the loop, as a function of time.

Problem 7.61 The magnetic field of an infinite straight wire carrying a steady current I can be obtained from the displacement current term in the Ampere/Maxwell law, as follows: Picture the current as consisting of a uniform line charge λmoving along the z axis at speed v (so that I=λ±¹), with a tiny gap of length E , which reaches the origin at time t=0. In the next instant (up to t=E/v) there is no real current passing through a circular Amperian loop in the xy plane, but there is a displacement current, due to the "missing" charge in the gap.

(a) Use Coulomb's law to calculate the z component of the electric field, for points in the xy plane a distances from the origin, due to a segment of wire with uniform density -λ . extending from toz1=vt-Etoz2=vt .

(b) Determine the flux of this electric field through a circle of radius a in the xy plane.

(c) Find the displacement current through this circle. Show thatId is equal to I , in the limit as the gap width (E)goes to zero.35

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