The focal length of a mirror is given by 1/f=1/u+1/v
where u and v represent object and image distances respectively,The maximum relative error in f is ?
f= uv/(u+v)
max error= depends on error in u, and v as compared to the true u,v.
To find the maximum relative error in f, we need to differentiate the given equation with respect to each variable (u and v) and then calculate the relative error.
Let's start by differentiating the equation with respect to u:
1/f = 1/u + 1/v
Differentiating both sides with respect to u:
d(1/f)/du = d(1/u)/du + d(1/v)/du
Since the derivative of a constant (1/f) is zero, we can simplify the equation:
0 = -1/u^2 + 0
Now, let's differentiate the equation with respect to v:
1/f = 1/u + 1/v
Differentiating both sides with respect to v:
d(1/f)/dv = d(1/u)/dv + d(1/v)/dv
Again, since the derivative of a constant (1/f) is zero, we can simplify the equation:
0 = 0 + (-1/v^2)
Now, let's solve these equations to find the relative error in f.
Using the given equation 1/f = 1/u + 1/v, we can rearrange it to solve for f:
1/f = (u + v)/(uv)
Now, we can substitute this expression into the equations we derived earlier to calculate the relative error:
Relative error in f with respect to u:
Error_u = (d(1/f)/du)/(1/f) = -1/u^2 / (1/u) = -1/u
Relative error in f with respect to v:
Error_v = (d(1/f)/dv)/(1/f) = (-1/v^2) / (1/u+v/uv) = -1/v
To find the maximum relative error, we need to consider the largest magnitudes of the relative errors. Therefore, the maximum relative error in f is the larger of the two absolute values:
Maximum relative error = max(abs(Error_u), abs(Error_v)) = max(abs(-1/u), abs(-1/v))
So, the maximum relative error in f is the larger of 1/abs(u) and 1/abs(v).