Two positively charged spheres, each with a charge of 2.0 * 10^-5...

Question

Two positively charged spheres, each with a charge of 2.0 * 10^-5 C, a mass of 1.0 kg, and separated by a distance of 1.0 cm, are held in place on a frictionless track. (b) If the spheres are released, will they move toward or away from each other? (c) What speed will each sphere attain as the distance between the spheres approaches infinity?

Accepted Answer

Step 1: Analyze the forces acting on the spheres. Since both spheres are positively charged, they will repel each other due to the electrostatic force. This repulsion will cause the spheres to move away from each other when released.. Step 2: Use Coulomb's Law to calculate the magnitude of the electrostatic force between the two spheres. Coulomb's Law is given by:

F = \frac{k q_{1} q_{2}}{r^{2}}

, where

k

is the Coulomb constant (8.99 × 10^9 N·m²/C²),

q_{1}

and

q_{2}

are the charges on the spheres (2.0 × 10^-5 C), and

r

is the distance between the spheres (1.0 cm or 0.01 m).. Step 3: Recognize that as the spheres move away from each other, the electrostatic potential energy is converted into kinetic energy. The total energy of the system is conserved. The initial potential energy is given by:

U = \frac{k q_{1} q_{2}}{r}

. As the distance approaches infinity, the potential energy becomes zero, and all the initial potential energy is converted into the kinetic energy of the two spheres.. Step 4: Write the expression for the kinetic energy of each sphere. Since the spheres have equal mass and charge, they will attain the same speed. The total kinetic energy of the system is given by:

K = 2 \frac{1}{2} m v^{2}

, where

m

is the mass of each sphere (1.0 kg) and

v

is the speed of each sphere. Set the total kinetic energy equal to the initial potential energy to solve for

v

.. Step 5: Solve for the speed

v

of each sphere. Rearrange the energy conservation equation to isolate

v

:

v = \sqrt{\frac{k q_{1} q_{2}}{m}}

. Substitute the known values for

k

,

q_{1}

,

q_{2}

,

m

, and

r

to calculate the final speed.

Key Concepts

Coulomb's Law

Conservation of Energy

Kinetic and Potential Energy