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Chapter 7: Q13P (page 316)

A square loop of wire, with sides of length a , lies in the first quadrant of the xy plane, with one comer at the origin. In this region, there is a nonuniform time-dependent magnetic field B(y,t)=ky3t2z^ (where k is a constant). Find the emf induced in the loop.

Short Answer

Expert verified

The induced emf into loop is -kt2a5.

Step by step solution

01

Write the given data from the question.

The side of the square loop is a.

The square loop exists in xy plane.

The non-uniform time dependent magnetic field,B(y,t)=ky3t2z^

02

Calculate the induced emf in the loop.

Let’s take small strip dy on the square loop which at distance y from the x-axis.


The magnetic flux is given by,

ϕ=B(y,t).dA

Here dA is the area of the strip.

Substitute Ky3t2for B(y,t)and ady for dA into above equation.

ϕ=ky3t2.ady

Integrate the above equation to calculate the flus through the entire loop.

ϕ=∫0aky3t2.adyϕ=kat2∫0ay3dyϕ=kat2y440a

Apply the limits,

ϕ=kat2a44-044ϕ=kat2a44ϕ=kt24a5

The induced emf in the loop is given by,

e=-dϕdt

Substitute kt24a5for ϕinto above equation.

e=-ddtkt24a5e=-2kt4a5e=-kt2a5

Hence the induced emf into loop is -kt2a5.

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Most popular questions from this chapter

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.

Question: A capacitor C has been charged up to potential V0at time t=0, it is connected to a resistor R, and begins to discharge (Fig. 7.5a).

(a) Determine the charge on the capacitor as a function of time,Q(t)What is the current through the resistor,l(t)?

(b) What was the original energy stored in the capacitor (Eq. 2.55)? By integrating Eq. 7.7, confirm that the heat delivered to the resistor is equal to the energy lost by the capacitor.

Now imagine charging up the capacitor, by connecting it (and the resistor) to a battery of voltage localid="1657603967769" V0, at time t = 0 (Fig. 7.5b).

(c) Again, determine localid="1657603955495" Q(t)and l(t).

(d) Find the total energy output of the battery (∫Vldt). Determine the heat delivered to the resistor. What is the final energy stored in the capacitor? What fraction of the work done by the battery shows up as energy in the capacitor? [Notice that the answer is independent of R!]

A circular wire loop (radius r , resistance R ) encloses a region of uniform magnetic field, B , perpendicular to its plane. The field (occupying the shaded region in Fig. 7.56) increases linearly with time(B=∝t)An ideal voltmeter (infinite internal resistance) is connected between points P and Q.

(a) What is the current in the loop?

(b) What does the voltmeter read? Answer:[∝r2/2]

An alternating current I(t)=I0cos(ωt) (amplitude 0.5 A, frequency ) flows down a straight wire, which runs along the axis of a toroidal coil with rectangular cross section (inner radius 1cm , outer radius 2 cm , height 1 cm, 1000 turns). The coil is connected to a 500Ω resistor.

(a) In the quasistatic approximation, what emf is induced in the toroid? Find the current, IR(t), in the resistor.

(b) Calculate the back emf in the coil, due to the current IR(t) . What is the ratio of the amplitudes of this back emf and the "direct" emf in (a)?

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=λv), 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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