A proton orbits a negatively charged wire that has a linear charge density of 8.0 x 10 -9 C/m. What speed must the proton have to maintain a circular orbit at 1.3 cm?
To find the speed of the proton required to maintain a circular orbit around the negatively charged wire, we can use the concepts of electrostatic force and centripetal force.
Step 1: Determine the electrostatic force between the proton and the negatively charged wire.
The electrostatic force between the proton and the negatively charged wire can be calculated using Coulomb's Law:
F = k * q1 * q2 / r^2,
where F is the electrostatic force, k is the electrostatic constant (8.99 x 10^9 N·m^2/C^2), q1 and q2 are the charges of the interacting objects (in this case, the charge of the proton and the charge density of the wire), and r is the distance between the charges (in this case, the radius of the circular orbit).
The charge of the proton is q1 = 1.6 x 10^-19 C.
The charge density of the wire is given as 8.0 x 10^-9 C/m. To convert this into charge per unit length for the circular orbit, we multiply by the circumference of the orbit to get:
q2 = (8.0 x 10^-9 C/m) * (2π * 1.3 cm).
Step 2: Calculate the radius of the circular orbit in meters.
Convert the radius of the circular orbit from centimeters to meters:
r = 1.3 cm = 1.3 x 10^-2 m.
Step 3: Calculate the electrostatic force between the proton and the wire.
Substitute the values into Coulomb's Law:
F = (8.99 x 10^9 N·m^2/C^2) * (1.6 x 10^-19 C) * (8.0 x 10^-9 C/m) * (2π * 1.3 x 10^-2 m)^2.
Step 4: Determine the centripetal force required for circular motion.
The electrostatic force between the proton and the wire will provide the centripetal force needed to maintain the circular orbit. Therefore, the centripetal force is equal to the electrostatic force:
F = mv^2 / r,
where m is the mass of the proton, v is its velocity, and r is the radius of the circular orbit.
Since the proton's mass is approximately 1.67 x 10^-27 kg, we can substitute the values:
mv^2 / r = (8.99 x 10^9 N·m^2/C^2) * (1.6 x 10^-19 C) * (8.0 x 10^-9 C/m) * (2π * 1.3 x 10^-2 m)^2.
Step 5: Solve for the velocity (v).
Rearrange the equation to isolate v:
v = √[(F * r) / (m)].
Substitute the values:
v = √[(8.99 x 10^9 N·m^2/C^2) * (1.6 x 10^-19 C) * (8.0 x 10^-9 C/m) * (2π * 1.3 x 10^-2 m)^2 / (1.67 x 10^-27 kg)].
To find the speed of the proton needed to maintain a circular orbit around the negatively charged wire, we can make use of the concepts of electrostatic forces and centripetal force.
Step 1: Determine the electrostatic force between the proton and the negatively charged wire.
The electrostatic force between the proton and the negatively charged wire is given by Coulomb's Law:
F = (k * q1 * q2) / r^2
Where F is the force, k is the electrostatic constant (9 x 10^9 N·m^2/C^2), q1 and q2 are the charges, and r is the distance between the charges.
In this scenario, q1 refers to the charge of the proton (1.6 x 10^-19 C) and q2 refers to the charge density of the wire (8.0 x 10^-9 C/m). Since the charge density is specified as per unit length, we need to multiply it by the circumference of the circular path (2πr) to get the total charge of the wire.
So the formula becomes:
F = (k * q1 * (q2 * 2πr)) / r^2
Step 2: Equate this electrostatic force to the centripetal force.
The centripetal force is given by:
F = (m * v^2) / r
Where m is the mass of the proton and v is its velocity or speed.
Since the electrostatic force and the centripetal force are the same, we can equate them:
(k * q1 * (q2 * 2πr)) / r^2 = (m * v^2) / r
Step 3: Solve for the speed (v).
Rearrange the equation to solve for v:
v^2 = ((k * q1 * (q2 * 2πr)) / (m * r))
v = √((k * q1 * (q2 * 2πr)) / (m * r))
Substituting the given values, we have:
v = √(((9 x 10^9 N·m^2/C^2) * (1.6 x 10^-19 C) * (8.0 x 10^-9 C/m) * (2π * 1.3 x 10^-2 m)) / ((1.67 x 10^-27 kg) * (1.3 x 10^-2 m)))
Simplifying this expression will give you the speed of the proton required to maintain a circular orbit around the negatively charged wire.