A compass needle free to turn in a horizontal plane is placed at the centre of circular coil of 30 turns and radius 12 cm. The coil is in a vertical plane making an angle of 45° with the magnetic meridian. When the current in the coil is 0.35 A, the needle points west to east.
(a) Determine the horizontal component of the earth's magnetic field at the location.
(b) The current in the coil is reversed, and the coil is rotated about its vertical axis by an angle of 90° in the anticlockwise sense looking from above. Predict the direction of the needle. Take the magnetic declination at the places to be zero.

Explanation

Here, plane of the coil is passing through de, vertical and perpendicular to plane of figure. Magnetic meridian is passing through ab and vertical also perpendicular to the plane of figure. `angle acd=435^@` is given.
Let `overset(to) (B_c ) = ` magnetic field at the centre of given coil due to current passing through it. `overset(to) (B_h)=` horizontal component of Earth.s magnetic field.
`overset(to) (B_R) = overset(to) (B_c) + overset(to)(B_h)=` resultant magnetic field at the centre `c` of given circular coil.
As per the statement, magnetic needle placed at the centre of coil, free to rotate in horizontal plane remains horizontal from west to east and so its magnetic dipole moment `overset(to) (m)` and resultant magnetic field `overset(to)(B_R)` both must be pointing from west to east. In this situation for the right angled `Delta CPQ`.
`sin45^@= (B_h)/( B_c)`
`therefore B_h= 0.7071 B_c`
`=0.7071 xx (mu_0)/( 2R)`
`=0.7071 xx ((4pi xx 10^(-7) )(30) (0.35) )/( (2) (0.12))`
`therefore B_h = 3.88 xx 10^(-5)` T
(b) Now, the coil is rotated by 90° about vertical axis passing through its plane and its centre, anticlockwise (as seen from top) then magnetic moment of the needle and resultant magnetic field `overset(to) (B_R)` both reverse their directions and once again needle comes in stable equilibrium condition. See the figure given below.

Here, `overset(to)(B._(R)) = overset(to)(B._(C ) ) + overset(to) (B_(h) ) " "(because overset(to) ( B_(h))= "constant" )`
Also, `overset(to) (m.) || overset(to) (B._(R) ) `