Here is my question:
Consider the bending of light by the gravitation of the Sun as described by Newtonian physics. Light of frequency f passes at a distance d from the center of the Sun, which has a mass M. Show that the bending angle of the light is proportional to M/d.
I'm absolutely stumped at the math process and steps in relating it to M/d. I'm sure I need to use Newton's Universal Gravitation equation somewhere, but how to get the end result is baffling to me. Any and all help is greatly appreciated.
I thought I answered this earlier today. Did you post the questiuon before?
http://www.jiskha.com/display.cgi?id=1211988677
To derive the relationship between the bending angle of light and the mass-to-distance ratio (M/d) of the Sun, we can start by using Newton's Universal Law of Gravitation and the equivalence between gravitational force and centripetal force.
Here's a step-by-step explanation of the math process:
Step 1: Start with the gravitational force equation.
According to Newton's Universal Law of Gravitation, the gravitational force between two objects is given by:
F = G * (m1 * m2) / r^2
Where F is the gravitational force, G is the gravitational constant, m1 and m2 are the masses of the two objects, and r is the distance between their centers of mass.
Step 2: Consider a light ray passing near the Sun.
In this scenario, we assume that the mass of the light ray is negligible compared to the Sun's mass. Therefore, we only need to consider the mass of the Sun, M.
Step 3: Apply the equivalence between gravitational force and centripetal force.
As the light ray passes near the Sun, it experiences a gravitational force towards the Sun's center. This force can be equated to the centripetal force, which keeps the light ray in a curved path.
Step 4: Express the centripetal force.
The centripetal force is given by:
F = m * v^2 / r
Where m is the mass of the object (in this case, the light ray), v is its velocity, and r is the radius of curvature of its path.
Step 5: Rearrange the equations.
Equating the gravitational force (from step 1) with the centripetal force (from step 4), we get:
G * (M * m) / d^2 = m * v^2 / r
Step 6: Eliminate the mass of the light ray (m).
Since the mass of the light ray is negligible, we can cancel it from the equation.
Step 7: Relate the velocity (v) to the speed of light (c).
In the case of light, the speed of light (c) is constant. Therefore, we have v = c.
Step 8: Express the radius of curvature (r) in terms of the bending angle (θ).
For a small bending angle, the radius of curvature can be approximated as:
r ≈ d / θ
Step 9: Substitute the expressions into the equation.
Substituting the expressions from steps 6, 7, and 8 into step 5, we get:
G * M / d^2 = c^2 / (d / θ)
Step 10: Rearrange the equation to solve for the bending angle (θ).
Rearranging the equation, we find:
θ = G * M / (c^2 * d)
Step 11: Simplify the equation.
Using the fact that G / c^2 is a very small quantity, we can approximate it as zero. Therefore, the equation simplifies to:
θ ≈ M / d
Thus, we obtain the relationship that the bending angle of light is approximately proportional to the mass-to-distance ratio (M/d).