Concave Mirrors: Videos & Practice Problems
Concave Mirrors are curved, reflective surfaces that bring parallel light to a focal point. For a spherical mirror, the focal length is set by the mirror’s curvature, with \(f=\frac{R}{2}\) . Because concave mirrors rely on reflection, their focal length does not depend on the surrounding medium’s index of refraction. Light from very distant objects is approximately parallel to the axis, so the image forms near the focal point.
Image location and image type are found with a principal ray diagram or the mirror equation, \( \frac{1}{d_o}+\frac{1}{d_i}=\frac{1}{f} \) . The main rays are: parallel then through \(F\), through \(F\) then parallel, and through \(C\) then back on itself. For objects beyond \(C\), the image is real, inverted, and reduced; between \(C\) and \(F\), it is real, inverted, and enlarged; inside \(F\), it is virtual, upright, and enlarged; at \(F\), the image is effectively at infinity.
Image size is described by magnification, \(M=\frac{h_i}{h_o}=-\frac{d_i}{d_o}\) . A negative magnification means inverted, a positive value means upright, a magnitude less than 1 means reduced, and a magnitude greater than 1 means enlarged.
Intro to Spherical Mirrors
You determine a concave mirror to have focal length . If you submerge the mirror in water (), what would be the new focal length?
Cannot be determined
A concave makeup mirror has a focal length of . What is the mirror’s radius of curvature?
Ray Diagrams for Concave Mirrors
For the following ray diagram, which ray is drawn incorrectly?
P-ray
F-ray
C-ray
None of the above.
An object is placed from a concave mirror, producing a real image from the mirror. What is the focal length of the mirror?
Ray Diagrams for Concave Mirrors Example 1
Magnification in Geometric Optics
A person stands in front of a large spherical mirror. A real image is formed in front of the mirror. What is the magnification of this image?
Magnification in Geometric Optics Example 2
Magnification in Geometric Optics Example 3
Ray Diagrams for Various Distances in Concave Mirrors
Draw the principal-ray diagram for an object placed exactly at the focal point of a concave mirror. Where does the image form?
An object is placed in front of a concave mirror. If a real, enlarged image is formed even farther from the mirror than the object, which of the following is the most likely value for the focal length of the mirror?
Ray Diagrams for Various Distances in Concave Mirrors Example 4
Ray Diagrams for Various Distances in Concave Mirrors Example 5
Here's what students ask on this topic:
In a concave mirror, the focal length () is directly related to the radius of curvature () of the mirror. The relationship is given by the formula . This means the focal length is exactly half the radius of curvature. The radius of curvature is the distance from the mirror's vertex to its center of curvature (), while the focal length is the distance from the vertex to the focal point (). This property is fundamental because it determines where parallel rays of light, such as those coming from distant objects, will converge after reflecting off the mirror.
The mirror equation relates the object distance (), image distance (), and focal length () of a concave mirror. It is expressed as . To find the image distance, you rearrange the equation to solve for : . Then take the reciprocal to get . Remember to apply the correct sign conventions: object distance is positive if the object is in front of the mirror (same side as incoming light), and focal length is positive for concave mirrors. This equation helps determine where the image forms relative to the mirror.
When drawing a principal ray diagram for a concave mirror, three main rays are typically used to locate the image:
- Parallel Ray (P-ray): This ray travels parallel to the principal axis from the top of the object and reflects through the focal point ().
- Focal Ray (F-ray): This ray passes through the focal point before hitting the mirror and reflects parallel to the principal axis.
- Center Ray (C-ray): This ray passes through the center of curvature () and reflects back on itself because it strikes the mirror at a right angle.
By drawing any two of these rays from the object's top and finding their intersection after reflection, you can determine the image's location, size, and orientation.
The position of the object relative to the focal point () and center of curvature () significantly influences the image's characteristics in a concave mirror:
- Object beyond : The image forms between and , is real, inverted, and reduced in size.
- Object between and : The image forms beyond , is real, inverted, and enlarged.
- Object inside : The image forms behind the mirror, is virtual, upright, and enlarged.
- Object at : The image forms at infinity, meaning the reflected rays are parallel and do not converge.
Understanding these positions helps predict image type, orientation, and size without complex calculations.
Magnification () in a concave mirror describes how much larger or smaller the image is compared to the object. It is calculated using the formula =
A negative magnification indicates the image is inverted, while a positive magnification means it is upright. If the absolute value of magnification is less than 1, the image is reduced; if greater than 1, the image is enlarged. This equation helps determine both the size and orientation of the image.
In concave mirrors, real images are formed when reflected rays actually converge at a point. These images can be projected onto a screen and are typically inverted. Real images occur when the object is placed beyond the focal point ().
Virtual images, on the other hand, occur when reflected rays diverge, and the brain traces them back to a point behind the mirror. These images cannot be projected onto a screen, appear upright, and are formed when the object is placed between the mirror and the focal point. Virtual images appear to be located behind the mirror.
Understanding this distinction is crucial for interpreting ray diagrams and solving mirror problems.
