A proton with speed v= 3 X 100000 m/sec orbits just outside a charged sphere of radius r = 1 cm . What is the charge on the sphere ?

To determine the charge on the sphere that allows a proton to orbit just outside it, we can use the principles of electrostatics and circular motion. The key here is to recognize that the electrostatic force acting on the proton provides the necessary centripetal force for its circular motion around the charged sphere.

Understanding the Forces Involved

When a charged particle like a proton orbits a charged sphere, it experiences an electrostatic force due to the charge on the sphere. This force can be described by Coulomb's law:

F = k * |q1 * q2| / r²

Where:

• F is the electrostatic force between the charges.
• k is Coulomb's constant, approximately 8.99 x 10^9 N m²/C².
• q1 is the charge of the sphere.
• q2 is the charge of the proton, which is approximately 1.6 x 10^-19 C.
• r is the distance from the center of the sphere to the proton, which is the radius of the sphere plus the distance just outside it (for simplicity, we can consider it as just the radius here).

Setting Up the Equation

The centripetal force required to keep the proton in circular motion is given by:

F_c = (m * v²) / r

Where:

• m is the mass of the proton, approximately 1.67 x 10^-27 kg.
• v is the speed of the proton, which is 3 x 10^5 m/s.
• r is the radius of the sphere, which is 0.01 m (1 cm).

Equating the Forces

For the proton to maintain its orbit, the electrostatic force must equal the centripetal force:

k * |q1 * q2| / r² = (m * v²) / r

Solving for the Charge

Rearranging the equation to solve for the charge on the sphere (q1), we get:

q1 = (m * v² * r) / (k * |q2|)

Now, substituting the known values:

• m = 1.67 x 10^-27 kg
• v = 3 x 10^5 m/s
• r = 0.01 m
• k = 8.99 x 10^9 N m²/C²
• q2 = 1.6 x 10^-19 C

Plugging these values into the equation:

q1 = (1.67 x 10^-27 kg * (3 x 10^5 m/s)² * 0.01 m) / (8.99 x 10^9 N m²/C² * 1.6 x 10^-19 C)

Calculating the numerator:

1.67 x 10^-27 kg * 9 x 10^{10} m²/s² * 0.01 m = 1.503 x 10^-18 kg m/s²

Now for the denominator:

8.99 x 10^9 N m²/C² * 1.6 x 10^-19 C = 1.4384 x 10^-9 N m²/C

Now, dividing the two results:

q1 = (1.503 x 10^-18) / (1.4384 x 10^-9) ≈ 1.04 x 10^-9 C

Final Result

The charge on the sphere is approximately 1.04 nC (nanocoulombs). This charge creates an electric field strong enough to keep the proton in orbit at the given speed and distance.