Q.Calculate the solid angle subtended by the periphery of an area of 1 cm$^2$ at a point situated symmetrically at a distance of 5 cm from the area.

The key idea is Solid Angle Calculation.

The solid angle $d\Omega$ subtended by a small area $dA$ at a point situated at a distance $r$ is given by the formula $d\Omega = \frac{dA \cos\theta}{r^2}$, where $\theta$ is the angle between the normal to the area and the line connecting the area to the point.

1. Given area $dA = 1 \text{ cm}^2$ and distance $r = 5 \text{ cm}$.
2. Since the point is situated symmetrically, the area is perpendicular to the line joining the point to the area's center, meaning $\theta = 0^\circ$. …

The concept of a solid angle is a three-dimensional analogue to the familiar two-dimensional angle. Just as a 2D angle measures the "spread" of lines originating from a point in a plane, a solid angle measures the "spread" of lines originating from a point in 3D space, forming a cone and subtending an area on a sphere centered at that point.

Imagine shining a flashlight onto a wall. The area illuminated on the wall depends on how far you are from it and how wide the beam is. The solid angle quantifies this "apparent size" of the illuminated area as seen from the flashlight's bulb.

The unit of solid angle is the steradian (sr). One steradian is the solid angle subtended at the center of a sphere by a portion of the surface whose area is equal to the square of the sphere's radius.

For a small planar area $dA$ subtending a solid angle $d\Omega$ at a point P at a distance $r$, the general formula is:

The problem states that the point is "situated symmetrically" from the area. This is a crucial piece of information. It means the point lies directly above the center of the area, along the normal to the area. In this specific configuration, the angle $\theta$ between the area's normal and the line connecting the area to the point is $0^\circ$. Since $\cos(0^\circ) = 1$, the formula simplifies significantly for this symmetrical case:

This simplified formula applies when the area is perpendicular to the line of sight from the point.

Now, let's calculate the solid angle step-by-step.

1. Identify the given parameters.

   • Area, $A = 1 \text{ cm}^2$
   • Distance, $r = 5 \text{ cm}$

2. Recall the appropriate formula for solid angle.

   Since the point is situated symmetrically, the simplified formula $\Omega = \frac{A}{r^2}$ is applicable.

3. Substitute the given values into the formula. …

Concept: Solid Angle and the Area-Squared Law

The solid angle $\Omega$ subtended by a small area $dA$ at a distance $r$ is given by:

where $\theta$ is the angle between the normal to the area and the line joining the point to the area.

Method: Direct Projection Method

This method is used when the area is small and the point is symmetrically placed.

Steps:

1. Identify the geometry

   The point is situated symmetrically at 5 cm from the area. This means the line from the point to the centre of the area is perpendicular to the area.

   Therefore, $\theta = 0^\circ$ and $\cos\theta = 1$.

2. Apply the solid angle formula

   For a small area $dA$ at perpendicular distance $r$:

3. Substitute the values
   • $dA = 1 \text{ cm}^2$
   • $r = 5 \text{ cm}$ …

Here are the common mistakes students make when solving this solid angle problem, along with how to avoid each.

1. Using the Wrong Formula for Solid Angle

The Mistake:

Students often try to use the formula for a point on the axis of a circular disc:

without checking if the area is actually a circle or if the point is on its axis. Here, the area is given as $1 \text{ cm}^2$, but its shape is not specified. The problem says the point is symmetrically situated, which means the point is on the normal through the center of the area, but the area could be a square, a rectangle, or any shape.

How to Avoid:

• Always check the geometry first. If the shape is not a circle, do not use the disc formula.
• For a small area $dA$ at a distance $r$ from the point, the exact formula is:

where $\theta$ is the angle between the normal to the area and the line joining the point to the area.

• Here, the point is symmetrically situated at 5 cm from the area, so the normal to the area points directly toward the point. Hence $\theta = 0^\circ$ and $\cos\theta = 1$.
• The correct approach: treat the area as a collection of infinitesimal elements, but since the distance (5 cm) is much larger than the size of the area ($\sqrt{1} = 1$ cm), we can approximate:

Key takeaway: Use $\Omega = A/r^2$ only when the area is small and perpendicular to the line of sight, and the point is far compared to the size of the area.

2. Forgetting the $\cos\theta$ Factor

The Mistake:

Even when using the correct formula $\Omega = A/r^2$, students forget that this is valid only when the area is perpendicular to the line joining the point to the area. If the area is tilted, the projected area $A\cos\theta$ must be used.

How to Avoid:

• Always draw a diagram. Mark the normal to the area and the line from the point to the center of the area.
• If the point is symmetrically situated, the normal points directly at the point, so $\theta = 0$ and $\cos\theta = 1$. No correction needed.
• If the problem had said "point is 5 cm away from the area but not on the normal," you would need to compute $\cos\theta$ and use:

Memory aid: The solid angle is the projected area divided by the square of the distance.

3. Confusing Solid Angle with Plane Angle

The Mistake:

Some students try to find a linear angle (like the half-angle $\theta$ of a cone) and then plug it into $\Omega = 2\pi(1-\cos\theta)$. This is unnecessary and often leads to errors if the area is not circular.

How to Avoid:

• Solid angle is a 3D concept (measured in steradians), not a 2D angle.
• If the area is small and flat, the simplest path is $\Omega = A_\perp / r^2$.
• Only use the cone formula if the problem explicitly describes a circular disc and the point is on its axis.

Quick check: If your answer comes out in degrees or radians, you’ve used the wrong formula.

4. Incorrect Unit Handling

The Mistake: …