A concave mirror magnifies an object placed 30.0cm from the mirror by a factor of +3.0. Calculate the radius of curvature of the mirror. Please, if possible include a step by step method. Thank you!
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Let's apply the mirror equation which is:
`1/d_o + 1/d_i=1/f`
where `d_o` ` `- object distance, `d_i` ` `- image distance and `f ` - focal length.
Note that, magnification factor is the ratio of image distance to object distance.
Magnification factor `= ` `-d_i/d_o`
Since magnification factor is +3 and object distance is 30 cm, then
`3=-d_i/30`
` `
Isolate `d_i` .
`-90=d_i`
` `
Then, substitute `d_o=30` cm and `d_i=-90` cm to the mirror equation.
`1/30 + 1/(-90)=1/f`
` `
To simplify the equation, multiply both sides by the LCD which is 90f.
`90f*(1/30 + 1/(-90))=(1/f) *90f`
`3f-f=90`
`2f=90`
Solve for f.
`f=90/2`
`f=45` cm
Moreover in curved mirrors, the focal length is half the raduis of the curvature of mirror. So,
`f = R/2`
Substitute value of f.
`45=R/2`
Isolate R.
`45*2 = R`
`90=R`
Hence, the raduis of curvature of the mirror is 90 cm.
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