An object is placed 10 cm in front of a convex mirror of focal length 4 cm. Find the image location by drawing a ray tracing diagram to scale. Verify your answer using the lens equation.

To find the image location using ray tracing, follow these steps:

Step 1: Draw a horizontal line representing the principal axis of the mirror.
Step 2: Mark the position of the convex mirror on the principal axis.
Step 3: Draw an arrow to represent the object, above the principal axis, at a distance of 10 cm in front of the mirror.
Step 4: Draw a ray from the top of the object, parallel to the principal axis. This ray will reflect through the focal point.
Step 5: Draw a ray from the top of the object, through the focal point. This ray will reflect parallel to the principal axis.
Step 6: Extend both rays behind the mirror.
Step 7: The point where the two extended rays intersect behind the mirror is the location of the image.

To verify the answer using the lens equation, we can use the following formula:

1/f = 1/do + 1/di

Where:
- f = focal length of the mirror
- do = object distance
- di = image distance

Plugging in the values:
1/4 = 1/10 + 1/di

Simplifying the equation:
1/di = 1/4 - 1/10
1/di = 5/20 - 2/20
1/di = 3/20
di = 20/3 cm

Therefore, the image location is 20/3 cm behind the mirror.

To find the image location using a ray tracing diagram, follow these steps:

1. Draw a horizontal line representing the principal axis of the convex mirror. Label it as the principal axis.

2. Mark the focal point (F) of the convex mirror. In this case, the focal length is 4 cm, so measure 4 cm from the mirror and mark it as the focal point.

3. Place an object on the principal axis, 10 cm in front of the mirror. Label it as the object.

4. Draw a ray parallel to the principal axis from the top of the object. After reflecting from the mirror, this ray should pass through the focal point.

5. Draw a ray from the top of the object towards the focal point of the mirror. After reflecting from the mirror, this ray should be parallel to the principal axis.

6. Extend both rays behind the mirror. The intersection point of these extended rays will be the location of the image. Label it as the image.

To verify the answer using the lens equation, we can use the formula:

1/f = 1/v - 1/u,

where:
f = focal length of the mirror (given as 4 cm),
v = image distance,
u = object distance.

In this problem, the object distance (u) is 10 cm in front of the mirror. Plugging these values into the equation:

1/4 = 1/v - 1/10.

To solve for v, we can rearrange the equation:

1/v = 1/4 + 1/10,
1/v = 14/40,
v = 40/14.

Simplifying further, we get:

v ≈ 2.86 cm.

Therefore, the image is located approximately 2.86 cm behind the mirror.