Question

Define focus of a concave spherical mirror. Prove that in a concave mirror \( R = 2f \).

Hint:

The focal length of a spherical mirror is half of its radius of curvature.

Answer 1

Definition of Focus:
The focus (\( F \)) of a concave mirror is the point on its principal axis where parallel rays of light converge after reflection from the mirror’s surface.
Proof that \( R = 2f \):
Consider a concave spherical mirror with:
- \( R \) as the radius of curvature
- \( C \) as the center of curvature
- \( P \) as the pole of the mirror
- \( F \) as the focal point
- \( f \) as the focal length
### Step 1: Relationship Between Radius of Curvature and Focal Length From the mirror formula:
\[ \frac{1}{f} = \frac{1}{u} + \frac{1}{v} \] For a parallel beam of light incident on a concave mirror:
- The incident rays are parallel, so \( u = \infty \).
- The reflected rays converge at the focus, so \( v = f \).
Thus, substituting in the mirror formula:
\[ \frac{1}{f} = \frac{1}{\infty} + \frac{1}{v} \] \[ \frac{1}{f} = \frac{1}{v} \] \[ v = f \] Since by definition, the center of curvature (\( C \)) lies at a distance \( R \) from the pole (\( P \)), we have the geometric relationship:
\[ R = 2f \] ### Conclusion:
Thus, the radius of curvature (\( R \)) of a concave mirror is twice its focal length (\( f \)), proving that:
\[ R = 2f \]