Animated Solution for Physics - Magnetic Effects of Current: There are two infinitely long straight current carrying conductors and they are held at right angles to each other so that their common ends meet at the origin as shown in the figure given below. The ratio of current in both conductors is 1 : 1. The magnetic field at point P is

Let's visualize the setup. We have two infinitely long wires meeting at the origin. Wire 1 lies along the $x$-axis, carrying current $I$ towards the right. Wire 2 lies along the $y$-axis, carrying the same current $I$ downwards towards the origin. We need to find the net magnetic field at point $P$, which has coordinates $(x,y)$.

To find the magnetic field at point $P$, we'll use the Biot-Savart Law for a straight current-carrying conductor. The magnetic field $B$ at a perpendicular distance $d$ is given by:

where $\alpha$ and $\beta$ are the angles subtended by the ends of the wire at the observation point.

Let's first focus on Wire 1. The perpendicular distance from point $P$ to Wire 1 is simply its $y$-coordinate. One end of the wire is at the origin, subtending an angle ${\theta }_1$, and the other end extends to infinity, subtending an angle of $90^{\circ }$. Also, using the right-hand thumb rule, the magnetic field at $P$ due to Wire 1 points outwards, out of the screen.

Now, look at the right-angled triangle formed by point $P$, the $x$-axis, and the origin. The sine of ${\theta }_1$ is the opposite side, $x$, divided by the hypotenuse, which is $\sqrt{x^2+y^2}$. Substituting this into our formula, we get the magnetic field $B_1$ due to Wire 1:

Next, let's analyze Wire 2. The perpendicular distance from point $P$ to Wire 2 is its $x$-coordinate. Similar to Wire 1, one end is at the origin, subtending an angle ${\theta }_2$, and the other end is at infinity, subtending $90^{\circ }$. Applying the right-hand thumb rule again, with the current flowing downwards, the magnetic field at $P$ also points outwards. This means the two fields will add up!

From the other right-angled triangle, the sine of ${\theta }_2$ is the opposite side, $y$, divided by the hypotenuse. Substituting this gives us the magnetic field $B_2$ due to Wire 2:

Since both magnetic fields point in the same outward direction, we simply add their magnitudes to find the total magnetic field at point $P$. Let's write down the sum of $B_1$ and $B_2$:

Now for the final algebraic simplification. We can take a common denominator of $xy$. The first terms combine to give $\frac{x+y}{xy}$. The second terms combine to give $x^2+y^2$ in the numerator, which partially cancels with the square root in the denominator.

Factoring out the common terms, we arrive at our final elegant expression for the total magnetic field:

This matches option (a) perfectly. Always pay close attention to the direction of current and use your right-hand rule carefully. Great job following along!