Final answer:
The mirror equation for a convex mirror relates the image and object distances to the focal distance. A convex mirror always produces a virtual image, independent of the location of the object.
Step-by-step explanation:
The mirror equation for a convex mirror is derived by considering a ray of light that is incident on the mirror and gets reflected. Let's consider an object situated at a distance 'u' from the mirror. The image formed by the mirror is at a distance 'v' from the mirror. The focal length of the convex mirror is denoted by 'f'. The mirror equation for a convex mirror is given by:
1/f = 1/v - 1/u
Now, to show that a convex mirror always produces a virtual image independent of the location of the object, we can consider two cases:
1. When the object is placed beyond the center of curvature of the convex mirror: In this case, the object distance 'u' is positive. Plugging the values of 'u' and 'f' into the mirror equation, we get a negative value for 'v', indicating that the image is virtual and located behind the mirror.
2. When the object is placed between the mirror and its center of curvature: In this scenario, the object distance 'u' is negative. Plugging the values of 'u' and 'f' into the mirror equation, we once again get a negative value for 'v', confirming that the image is virtual and located behind the mirror.
Hence, regardless of the object's position, a convex mirror always produces a virtual image.