There is a ring of radius r and charge Q on it wats the increase in its tension if charge q is put at center?

The force acting on the charge q due to charge Q of the ring is:

F = kQq/R²

This is also the net force applied on the ring as well due to charge q on Q.

However, the stress force that the ring feels would be:

F/L = (kQq/R²) / 2πR = kQq/2πR³

Nice question. But note that the tension will decrease if the central charge is of the opposite sign.

In any case, there are two parts to the problem:

1. Determining the radial force per unit angle

2. Computing the tension resulting from a given radial force.

The first part is very simple: for each differential angle element d@, the force on the ring associated with that differential element is: k(q)(Qr)/(r^2) where Qr is the differential charge element and is equal to Q/(2pi)

The second part is more interesting. Suppose we were to cut the ring in half on a diameter. Then each half would experience a force due to the central charge and start moving. This force is what the "tension" in the ring had been opposing, so the "tension" in the ring must equal half the force on each half.

Since each half is symmetrical, the force is along the axis of symmetry, so we only have to concern ourselves with that component of the force.

If we place our coordinate system with the origin at the center of the ring and the cut along the X axis, the axis of symmetry is the Y axis and the force on the top half of the ring is:

Ft = integral from @ = 0 to pi of Fy(@)d@

where Fy(@) = the y component of the uniform radial force F

= F sin @

Since F (the radial force due to the central charge) is uniform, we can pull it out of the integral to get:

Ft = F integral for 0 to pi of sin@ d@

This force is supported by the two ends of the half ring so:

2T = Ft

Note that the second part of the problem applies not only to the ring of charge but to other interesting problems such as how strong a hoop must be in order to withstand spinning at a specified angular velocity.

the opportunity of a hoop of cost could be got here upon by skill of superposing the point cost potentials of infinitesmal cost factors. it really is an celebration of a continuous cost distribution. the hoop means can then be used as a cost ingredient to calculate the opportunity of a charged disc. because the potential is a scalar volume, and considering that each and every ingredient of the hoop is an similar distance r from the point P, the potential is in simple terms given by skill of If the cost is characterised by skill of a community density and the hoop by skill of an incremental width dR', then: