Final answer:
The image by refraction at a concave glass surface forms at -7.5 cm from the vertex on the same side as the object, indicating that it's a virtual image. The magnification is 0.25, meaning that the image is a quarter of the object's size.
Step-by-step explanation:
To find the location of the image formed by refraction at a concave surface and its magnification, we use the lensmaker's formula and magnification equations. Given that the object is located in air (with index of refraction nair = 1) 30 cm from the vertex of a concave glass surface (with index of refraction nglass = 1.5) and the radius of curvature of the surface is 10 cm, we can calculate the image distance and magnification.
Step 1: Convert the radius of curvature
For a concave surface, the radius of curvature R is considered negative, so R = -10 cm.
Step 2: Use the lensmaker's formula
The lensmaker's formula for refraction at a spherical surface is given by (n2 - n1) / R = (n2 / v) - (n1 / u), where n1 is the refractive index of the initial medium (air), n2 is the refractive index of the glass, v is the image distance, and u is the object distance (which is positive if the object is on the same side as the incoming light).
Step 3: Solve for image distance (v)
Substituting into the formula: (1.5 - 1) / -10 = (1.5 / v) - (1 / -30), we find that the image distance v is approximately -7.5 cm. The negative sign indicates that the image is formed on the same side as the object (virtual image).
Step 4: Calculate magnification (m)
Magnification m is given by the ratio of image distance to object distance (m = -v/u). Substituting the values, we get m = -(-7.5) / 30, which simplifies to m = 0.25. This magnification indicates that the image is 1/4th the size of the object.
Therefore, the image by refraction forms at -7.5 cm from the vertex on the same side as the object with a magnification of 0.25.