Question

A magnetic needle free to rotate in a vertical plane parallel to the magnetic meridian has its North tip pointing down at \[22^\circ \] with horizontal. The horizontal components of the earth’s magnetic field at the place is known to be \[0.35G\]. The magnetic of the earth’s magnetic field at the place will be:
(A) \[0.68G\]
(B) \[0.36G\]
(C) \[0.83G\]
(D) \[0.86G\]

Answer 1

Hint The direction of the north tip of the needle is the direction of the magnetic field vector. The component of the magnetic field vector on the horizontal axis will give the magnitude of the magnetic field multiplied by the angle it makes with the horizontal.
Formula used: In this solution we will be using the following formulae;
\[{V_A} = V\cos \theta \] where \[{V_x}\] is the component of a vector on an arbitrary axis A, \[V\] is the magnitude of the vector, and \[\theta \] is the angle between the vector and the A axis.

Complete Step-by-Step solution:
According to the question, a magnetic needle free to rotate vertically (no resistance to motion on vertical axis) is pointing at 22 degree angle to the horizontal.
Generally, the direction of the magnetic field is the direction of the north tip of the needle. Hence, this means that the magnetic field vector makes an angle 22 degrees with the horizontal. Hence, the component of the magnetic vector to the horizontal gives the known horizontal component as mentioned in the question.
The component of a vector to any axis can be given as
\[{V_A} = V\cos \theta \] where \[{V_x}\] is the component of a vector on an arbitrary axis A, \[V\] is the magnitude of the vector, and \[\theta \] is the angle between the vector and the A axis.
Hence, for magnetic field on axis x, we have
\[{B_x} = B\cos \theta \]
\[ \Rightarrow 0.35G = B\cos 22^\circ \]
Hence,
\[B = \dfrac{{0.35G}}{{\cos 22}} = 0.38G\]
Answers are not given in options.

Note: For clarity, note that the north tip of the needle actually points to the magnetic south pole of the earth. But since, the direction of magnetic field is defined to point from north to south, then the tip points in the direction of magnetic field too (southward).