A 6-μg (6 × 10−9kg) -particle carrying 2 nC charge describes a circular orbit about an in- finitely long and thin charged wire.
(a) What is the sign of the charge on the wire? The particle charge’s sign is positive.
(b) What is the line charge density of the wire if the speed of the particle is 100 m/s
To find the line charge density of the wire, we can use the formula for the centripetal force:
F = (m*v^2) / r
where F is the electrostatic force on the particle due to the charged wire, m is the mass of the particle, v is the speed of the particle, and r is the radius of the circular orbit.
The electrostatic force between the particle and the wire can be determined using Coulomb's law:
F = k * (q1 * q2) / r^2
where k is the Coulomb constant, q1 is the charge on the particle, q2 is the charge on the wire, and r is the distance between the particle and the wire.
Setting these two formulas equal to each other, we get:
k * (q1 * q2) / r^2 = (m*v^2) / r
Now we can plug in the values given:
k = 9 × 10^9 N m^2/C^2
q1 = 2 nC = 2 × 10^-9 C
m = 6 × 10^-9 kg
v = 100 m/s
Plugging in these values and solving for q2, we get:
(9 × 10^9) * (2 × 10^-9) * q2 / r^2 = (6 × 10^-9) * (100)^2 / r
Solving for q2, we find:
q2 = (6 × 10^-9) * (100)^2 / ((9 × 10^9) * (2 × 10^-9)) = 333 × 10^-9 C
The line charge density of the wire can be calculated using:
λ = q / L
where q is the total charge on the wire and L is the length of the wire. Assuming the wire is infinitely long, the length L can be considered very large.
Therefore, the line charge density of the wire is:
λ = 333 × 10^-9 C / L
This is the line charge density of the wire in C/m.