An object is located 1.35 m from an 8.0-D lens. By how much does the image move if the object is moved (a) 0.90 m closer to the lens, and (b) 0.90 m farther from the lens?

Verified step by step guidance

1. Step 1: Recall the lens formula: $\frac{1}{f}=\frac{1}{d}$_o + 1d_i, where $f$ is the focal length of the lens, $d$_o is the object distance, and $d$_i is the image distance. The focal length $f$ can be calculated from the lens power $P$ using the formula $f=\frac{1}{P}$. Substitute $P=\; 8.0$ diopters to find $f$ in meters.
2. Step 2: Using the lens formula, calculate the initial image distance $d$_i when the object is located at $d$_o = 1.35 m. Rearrange the lens formula to solve for $d$_i: $d$_i = 11f - 1d_o. Substitute the values of $f$ and $d$_o to find $d$_i.
3. Step 3: For part (a), calculate the new object distance when the object is moved 0.90 m closer to the lens. The new object distance is $d$_o = 1.35 - 0.90 m. Use the lens formula again to calculate the new image distance $d$_i for this new object distance.
4. Step 4: For part (b), calculate the new object distance when the object is moved 0.90 m farther from the lens. The new object distance is $d$_o = 1.35 + 0.90 m. Use the lens formula again to calculate the new image distance $d$_i for this new object distance.
5. Step 5: To find the image movement in each case, subtract the initial image distance from the new image distance for both part (a) and part (b). The magnitude of the difference gives the amount by which the image moves in each scenario.

Key Concepts

Lens Formula

The lens formula relates the object distance (u), image distance (v), and focal length (f) of a lens through the equation 1/f = 1/v - 1/u. This formula is essential for determining the position of the image formed by a lens based on the position of the object.

Magnification

Magnification (M) is the ratio of the height of the image (h') to the height of the object (h), and it can also be expressed as M = -v/u. Understanding magnification helps in analyzing how the size and position of the image change as the object is moved relative to the lens.

Sign Convention for Lenses

The sign convention for lenses dictates that distances measured in the direction of the incoming light (toward the lens) are negative, while distances measured in the direction of the outgoing light (away from the lens) are positive. This convention is crucial for correctly applying the lens formula and interpreting the results.