Question

deduce the inverse square law. F proportional to 1/r square ​

Answer 1

F is proportional to 'product of charges'

F is inversely proportional to '(separation)square'

hence F is proportional to

(q1 × q2) / (r)square

hence F = (kq1 q2) / (r)square

where k is the proportionality constant.

hope it helps

Answer 2

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Consider a circular orbit (Kepler's first law tells us this is possible, as circles are particular cases of ellipses). By Kepler's second law, the speed v is constant along the orbit. We can obtain its dependence on r using Kepler's third law: T2∝r3.

The result is v2∝1/r.

For the orbit to be circular, the force should satisfy...

F(r)=mv2r∝1r2.

We can also find the direction of the force from Kepler's laws! We work in two dimensions because Kepler's first law tell us that the orbits stay in a plane. The acceleration in radial coordinates is

a⃗ =(r¨−rθ˙2)r^+(rθ¨+2r˙θ˙)θ^.

Notice that the θ^ component of the acceleration is just 1rddt(r2θ˙), and that r2θ˙ is the areal velocity, which is constant by Kepler's second law. Therefore, the acceleration has the direction of r^, and so does the force. The latter should then be of the form

F⃗ (r⃗ )=−F(r)r^ ,

where the dependence only on r and not θ is a consequence of the isotropy of space.

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