A convex mirror with a radius of curvature of 45.0 cm forms a 1.70 cm tall image of a pencil at

A convex mirror with a radius of curvature of 45.0 cm forms...

A convex mirror with a radius of curvature of $45.0 \mathrm{cm}$ forms a $1.70 \mathrm{cm}$ tall image of a pencil at a distance of $15.8 \mathrm{cm}$ behind the mirror. Calculate the object distance for the pencil and its height. Is the image real or virtual? What is the magnification? Is the image inverted or upright?

Key Concepts

Mirror Equation

The mirror equation, 1/f = 1/d_o + 1/d_i, is a fundamental relation in geometrical optics that relates the focal length (f) of a spherical mirror to the distances of the object (d_o) and the image (d_i) from the mirror. This relation is essential for solving problems where any two of these distances are known and the third must be found.

Sign Conventions

In mirror problems, particularly with curved surfaces like convex and concave mirrors, sign conventions are critical for obtaining the correct result. They determine the positive or negative values for the focal length, object distance, and image distance based on the orientation and nature (real or virtual) of the image, which is paramount for correct application of the mirror equation and the magnification formula.

Convex Mirror Characteristics

Convex mirrors are curved outward and always form virtual, upright, and reduced images regardless of the object position. Their inherent geometry leads to a focal length that, according to the chosen sign convention, is treated with a particular sign (often negative in the common sign convention), and these mirrors are extensively used in applications where a wider field of view is beneficial.

Magnification

Magnification in mirror optics is defined as the ratio of the image height to the object height, which can also be expressed as m = -d_i/d_o. It indicates not only the relative size of the image compared to the object but also whether the image is inverted (negative magnification) or upright (positive magnification). This concept is crucial for understanding how the image will appear relative to the actual object.

Transcript

00:01 In this problem, we are given a convex mirror and we are given some information about the image.

00:07 We have the position of the image to be 15 .8 centimeters to the right, which is why it has a negative sign, and we have the image's height of 1 .7 centimeters.

00:18 We are also told the radius of curvature for the mirror is 45 centimeters.

00:23 Using this equation for a mirror with radius r, we can calculate the value of p given q and r first we solve for p in this equation we will have one over p equals negative two over r minus one over q solving for p we take the inverse of both sides or reciprocal and we can have that negative two over r minus one over q to the minus one power is equal to p which is the distance to the object plugging the values in for r equals forty four centimeters and q equals negative 15 .8 centimeters.

01:12 Plug this into a calculator gives us a value of p equals 53 .1 centimeters or just 53 centimeters with figs.

01:28 Next we would like to determine the magnification of the system.

01:34 In order to do this we can use one of the formulas m equals negative q over p.

01:41 So we use the ratio between the distances of the image and object to the mirror.

01:47 We have q and we have just calculated p.

01:50 So we can plug in these values and we have a positive 15 .8 centimeters and the numerator be the two negatives cancel and a 53 centimeters in the denominator.

02:03 This tells us the total magnification will be 0 .298.

02:12 We can round this to 0 .30 if we want to use the two sig figs.

02:19 I will just keep this extra figure because we will be using this number in the next calculation...

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