Magnetic Flux and Flux Density

Magnetic Flux and Flux Density

• The magnetic flux density   is analogous to the electric flux density  The relation between   and  is already mentioned, which is through the property of medium called permeability u. The relation is given by,

(7.11.1)

For the free space, μ = μ0 = 4л × 10^-7 Н/m hence,

• The magnetic flux density has units Wb/m² and hence it can be defined as the flux in webers passing through unit area in a plane at right angles to the direction flux.

• If the flux passing through the unit area is not exactly at right angles to the plane consisting the area but making some angle with the plane then the flux crossing the area is given by,

• Now consider a closed surface which is defining a certain volume. The magnetic flux lines are always exist in the form of closed loop. Thus for a closed surface the number of magnetic flux lines entering must be equal to the number of magnetic flux lines leaving. The single magnetic pole can not exists like a single isolated electric charge. No magnetic flux can reside in a closed surface. Hence the integral  evaluated over a closed surface is always zero.

• This is called law of conservation of magnetic flux or Gauss's law in integral form for magnetic fields.

• Applying divergence theorem to equation (7.11.4),

• The divergence of magnetic flux density is always zero. This is called Gauss's law in differential form for magnetic fields. This is another Maxwell's equation.

1. Maxwell's Equations for Static Electromagnetic Fields

• Let us summarize the Maxwell's equations for static electric and magnetic fields.

• The Maxwell's equations in integral form can be summarized as,

2. Application of Flux Density and Flux to Co-axial Cable

• Let us obtain the flux between the conductors of a co-axial cable using the concepts of flux density and the flux.

• The co-axial cable is shown in the Fig. 7.11.1, such that its axis is along the z-axis.

The radius of the inner conductor is 'a' while the inner radius of the outer conductor is 'b'. It carries a direct current I which is uniformly distributed in the inner conductor. The outer conductor carries same current I in opposite direction to that carried by the inner conductor.

• As derived in the section 7.9.2,  in the region a < r < b is given by,

• We are interested in the flux in the region a < r < b. The cable is filled with the air as dielectric with µ = µ0.

• Let d be the length of the conductors. The magnetic flux contained between the conductors in a length d is the magnetic flux crossing the radial plane from r = a to r = b and for z = 0 to z = d.

The magnetic flux is given by,

Ex. 7.11.1 In cylindrical co-ordinates  Tesla. Determine the magnetic flux A crossing the plane surface defined by 0.5 ≤ r ≤ 2.5m and 0 ≤ z ≤ 2 m

Sol. : The surface is shown in the Fig. 7.11.2.

Ex. 7.11.2 Find the flux passing the portion of the plane ϕ = π/ 4 defined by

0.01 <  r < 0.05 m and 0 < z <  2m A current filament of 2.5 A is along the z axis in the  direction, in free space.

Sol. : The arrangement is shown in the Fig. 7.11.3.

Due to current carrying conductor in free space along z axis,  is given by,

The flux crossing the surface is given by,

Ex. 7.11.3 Derive a general expression for the magnetic flux density B at any point along the axis of a long solenoid. Sketch the variation of B from point to point along the axis.

AU : May-05, 13, Dec.-13, 14, Marks 16

Sol. : Consider the cross-section of the solenoid as shown in the Fig. 7.11.4 with its axis along z direction.

The solenoid is made up of turns which are arranged in circular loops. Thus a circular loop at dz produces a magnetic field at point P which is at a distance z on its axis. Let current through solenoid is I amperes.

Use the result of section 7.7 for obtaining H on the axis of a circular loop with r = a.

Let N / L = Number of turns per unit length.

The variation B along the axis is shown in the Fig. 7.11.5.

It can be seen that the field strength at the end of the long solenoid is one half of that at the center.

Examples for Practice

Ex. 7.11.4 A radial field,  exists in free space. Find the magnetic flux crossing the surface defined by 0 ≤ ϕ ≤ π/ and 0 <z < 1 m.

[Ans.: 2.1236 Wb]

Ex. 7.11.5 A radial magnectic field  exists in free space.

Find the magnetic flux ϕ crossing the surface defined by – π/4 ≤ ϕ π/4, 0 ≤ z ≤ 1m.

 [Ans. : 3.9789 Wb]

Review Questions

1. Derive the expression for the flux in a coaxial cable.

2. Write a note on magnetic flux density.

3. What is law of conservation of magnetic flux ?

4. Show that the magnetic field intensity at the end of a long solenoid is one half of that at the center.