# Physics Class 12 NCERT Solutions: Chapter 4 Moving Charges and Magnetism Part 5

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Q: 15. A magnetic field of is required which is uniform in a region of linear dimension about 10 cm and area of cross-section about . The maximum current-carrying capacity of a given coil of wire is and the number of turns per unit length that can be wound round a core is at most turns. Suggest some appropriate design particulars of a solenoid for the required purpose. Assume the core is not ferromagnetic

Magnetic field strength,

Number of turns per unit length,

Current flowing in the coil,

Permeability of free space,

Magnetic field is given by the relation,

If the length of the coil is taken as, radius, number of turns, and current, then these values are not unique for the given purpose. There is always a possibility of some adjustments with limits.

Q: 16. For a circular coil of radius and turns carrying current I, the magnitude of the magnetic field at a point on its axis at a distance x from its centre is given by,

(A) Show that this reduces to the familiar result for field at the centre of the coil.

(B) Consider two parallel co-axial circular coils of equal radius, and number of turns, carrying equal currents in the same direction, and separated by a distance. Show that the field on the axis around the mid-point between the coils is uniform over a distance that is small as compared to , and is given by, , approximately. [Such an arrangement to produce a nearly uniform magnetic field over a small region is known as Helmholtz coils.]

Number of turns on the coil

Current in the coil

Magnetic field at a point on its axis at distance x is given by the relation,

Where,

Permeability of free space

(A) If the magnetic field at the centre of the coil is considered, then.

This is the familiar result for magnetic field at the centre of the coil.

(B) Radii of two parallel co-axial circular coils

Number of turns on each coil

Current in both coils

Distance between both the coils

Let us consider point Q at distance d from the centre.

Then, one coil is at a distance of from point Q.

Magnetic field at point Q is given as:

Also, the other coil is at a distance of from point.

Magnetic field due to this coil is given as:

Total magnetic field,

For, neglecting the factor, we get:

Hence, it is proved that the field on the axis around the mid-point between the coils is uniform.

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