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Chapter 5: Q5P (page 222)

A current Iflows down a wire of radius a.

(a) If it is uniformly distributed over the surface, what is the surface current density K?

(b) If it is distributed in such a way that the volume current density is inversely

proportional to the distance from the axis, what is J(s)?

Short Answer

Expert verified

(a) The surface current density corresponding to a current Iflowing down a wire of radius aisI2πa.

(b) The volume current density corresponding to a current Iflowing down a wire of radius ais I2πas.

Step by step solution

01

Given data

There is a wire of radius a carrying a current I.

02

Surface and volume current density

The surface current density if the current is uniformly distributed over a surface is

K=IL.....(1)

Here, L is the cross sectional length of the surface.

The total current in terms of volume current density is

I=∫Jda.....(2)

Here, the integration is over the cross sectional area.

03

Surface current density in wire of radius   

From equation (1), the surface current density on the surface of the wire of radius a is

K=I2πa

Thus, the uniform surface current density is I2πa.

04

Volume current density in wire of radius  

Let the volume current density be

J=ks.....3

Here, k is a constant and is the radial component of polar coordinate.

From equation (2),

I=k∫0a1s2πsds=k2πak=I2πa

Substitute this in equation (3) to get

J=I2πas

Thus, the volume current density is I2πas.

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Most popular questions from this chapter

A uniformly charged solid sphere of radius Rcarries a total charge Q, and is set spinning with angular velocitywabout the zaxis.

(a) What is the magnetic dipole moment of the sphere?

(b) Find the average magnetic field within the sphere (see Prob. 5.59).

(c) Find the approximate vector potential at a point (r,B)where r>R.

(d) Find the exact potential at a point (r,B)outside the sphere, and check that it is consistent with (c). [Hint: refer to Ex. 5.11.]

(e) Find the magnetic field at a point (r, B) inside the sphere (Prob. 5.30), and check that it is consistent with (b).

The magnetic field on the axis of a circular current loop (Eq. 5.41) is far from uniform (it falls off sharply with increasing z). You can produce a more nearly uniform field by using two such loops a distanced apart (Fig. 5.59).

(a) Find the field (B) as a function of \(z\), and show that \(\frac{\partial \mathbf{B}}{\partial \mathbf{z}}\) is zero at the point midway between them \((z=0)\)

(b) If you pick d just right, the second derivative of \(B\) will also vanish at the midpoint. This arrangement is known as a Helmholtz coil; it's a convenient way of producing relatively uniform fields in the laboratory. Determine \(d\) such that

\(\partial^{2} B / \partial z^{2}=0\) at the midpoint, and find the resulting magnetic field at the center.

\(\frac{A I_{0}}{5 \sqrt{5} R}\)

Question: (a) Find the density ρof mobile charges in a piece of copper, assuming each atom contributes one free electron. [Look up the necessary physical constants.]

(b) Calculate the average electron velocity in a copper wire 1 mm in diameter, carrying a current of 1 A. [Note:This is literally a snail'space. How, then, can you carry on a long distance telephone conversation?]

(c) What is the force of attraction between two such wires, 1 em apart?

(d) If you could somehow remove the stationary positive charges, what would the electrical repulsion force be? How many times greater than the magnetic force is it?

Find and sketch the trajectory of the particle in Ex. 5.2, if it starts at

the origin with velocity

(a)v→(0)=EBy⏜(b)v→(0)=E2By⏜(c)v→(0)=EB(y⏜+z⏜).

A steady current Iflows down a long cylindrical wire of radius a(Fig. 5.40). Find the magnetic field, both inside and outside the wire, if

  1. The current is uniformly distributed over the outside surface of the wire.
  2. The current is distributed in such a way that Jis proportional to s,the distance from the axis.
See all solutions

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