Consider a long charged straight wire that lies fixed and a particle of charge +2e and mass 6.70E-27 kg. When the particle is at a distance 1.31 cm from the wire it has a speed 2.30E+5 m/s, going away from the wire. When it is at a new distance of 2.71 cm, its speed is 3.10E+6 m/s. What is the charge density of the wire?

Calculate the potential which is energy per unit charge (voltage) .0131 meters from a wire of charge density s. (assuming voltage 0 at infinity)

Then calculate the voltage .0271 meters away.
change in potential energy of the particle in moving is the charge times the change in voltage
that is the change in kinetic energy (1/2 mv^2 if low compared to light speed)

To find the charge density of the wire, we can use the concept of electric field and the centripetal force experienced by the particle.

Step 1: Determine the electric field at the initial distance from the wire.
We know that the electric field created by a long straight wire is given by the equation: E = (λ / 2πε0r), where λ is the linear charge density, ε0 is the permittivity of free space, and r is the distance from the wire.
At a distance of 1.31 cm (or 0.0131 m) from the wire, the electric field is given by:
E1 = (λ / 2πε0 * 0.0131)

Step 2: Calculate the force experienced by the particle at the initial distance.
The force experienced by a charged particle moving in an electric field is given by the equation: F = qE, where q is the charge of the particle.
The force acting on the particle is the centripetal force, which is given by: F = (mv^2) / r, where m is the mass of the particle, and v is the velocity.
At the initial distance, the force experienced by the particle is:
F1 = (mv^2) / r

Step 3: Set the electric force equal to the centripetal force and solve for the charge density.
As the particle is moving away from the wire, the electric force and the centripetal force are in opposite directions. Therefore, we can set up the equation: F1 + qE1 = 0. This implies that:
(mv^2) / r + q * (λ / 2πε0 * 0.0131) = 0

Step 4: Calculate the charge density (λ).
Solving the equation from step 3 for λ gives us the charge density of the wire.

Keep in mind that q = +2e, where e is the charge of an electron (+1.6 x 10^-19 C), m = 6.70 x 10^-27 kg, v = 2.30 x 10^5 m/s, and r = 0.0131 m.

Evaluating the equation will give you the charge density of the wire.