Animated Solution for Physics - Magnetic Effects of Current: Choose the correct option.

Imagine you are standing on the Earth's surface, holding a magnetic compass. The vertical plane that perfectly aligns with the magnetic north and south poles is called the Magnetic Meridian. If you allow a magnetic needle to pivot vertically in this plane, the angle it makes with the horizontal is the true dip ($\rho$).

But what happens if you force the needle to pivot in a different vertical plane? Let's say this new plane is at an angle $\alpha$ to the magnetic meridian. The angle the needle makes with the horizontal in this new plane is called the apparent dip (${\rho }^{\prime }$).

To understand the relationship between true and apparent dip, we need to look at the components of the Earth's magnetic field. When we shift to a new vertical plane, the vertical component of the magnetic field ($B_V$) remains exactly the same. However, the horizontal component ($B_H$) changes because we are only looking at its projection in the new plane.

The new horizontal component is given by:

Now, let's write the equations for the dip angles. For the true dip, the tangent of the angle is the ratio of the vertical to the horizontal component:

Similarly, for the apparent dip:

Let's substitute our known values into the apparent dip equation. Since $B_V^{\prime }=B_V$ and $B_H^{\prime }=B_H\cos \alpha$, we get:

We can rewrite this using our true dip equation:

Here is the crucial mathematical catch: the value of $\cos \alpha$ is always less than or equal to $1$. This means that $\frac{1}{\cos \alpha }$ is always greater than or equal to $1$. Therefore:

Since the tangent function is strictly increasing for angles between $0^{\circ }$ and $90^{\circ }$, it directly implies that:

This means the true dip is always less than or equal to the apparent dip. The correct option is (b).