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?
To find the charge density of the wire, we'll use the concept of electric field and gravitational force acting on the particle.
Step 1: Find the electric field at the initial position of the particle.
The electric field produced by a long charged wire is given by the formula:
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.
Given that the initial distance from the wire is 1.31 cm = 0.0131 m, and the electric field is given by:
E₁ = (λ / 2πε₀r₁)
Step 2: Find the electric field at the final position of the particle.
The new distance from the wire is 2.71 cm = 0.0271 m, and the electric field is given by:
E₂ = (λ / 2πε₀r₂)
Step 3: Use the concept of electric field and gravitational force to determine the charge density.
The electric field exerts a force on a charged particle given by:
F = qE
where q is the charge of the particle.
The gravitational force acting on the particle is given by:
F = mg
where m is the mass of the particle and g is the acceleration due to gravity.
Given that the particle has a charge of +2e and a mass of 6.70E-27 kg, and the gravitational force is given by:
F = (2e)E₁ = mg
Step 4: Solve for the linear charge density.
Setting the two equations for force equal to each other, we have:
(2e)E₁ = mg
Substituting the expressions for electric field and gravitational force:
(2e)(λ / 2πε₀r₁) = mg
Simplifying, we get:
λ = (4πε₀r₁mg) / (2e)
Let's calculate the value using the given data:
ε₀ = 8.854E-12 C²/(N*m²), r₁ = 0.0131 m, m = 6.70E-27 kg, and e = 1.60E-19 C.
λ = (4π * 8.854E-12 C²/(N*m²) * 0.0131 m * 6.70E-27 kg * 9.81 m/s²) / (2 * 1.60E-19 C)
λ ≈ 1.30E-5 C/m
Hence, the charge density of the wire is approximately 1.30E-5 C/m.
To find the charge density of the wire, we need to use the equations of motion and the electric force between the wire and the charged particle.
The equation to calculate the electric force between a wire and a charged particle is given by:
F = k * (Q1 * Q2) / r^2
Where F is the electric force, k is the electrostatic constant (9 x 10^9 N m^2/C^2), Q1 and Q2 are the charges of the wire and particle respectively, and r is the distance between them.
In this case, the charge of the particle is +2e = 2 * 1.6 x 10^-19 C = 3.2 x 10^-19 C.
We can assume that the wire has an infinite length, so it will have a uniform charge density.
The linear charge density (λ) is given by:
λ = Q / L
Where Q is the total charge on the wire and L is the length of the wire.
Since we are given the mass of the particle, we can use the equation for centripetal force to relate the electric force to the centripetal force.
The centripetal force is given by:
F = (m * v^2) / r
Where m is the mass of the particle, v is its velocity, and r is the radius of the circular path.
Now, we can use the given information to solve for the charge density of the wire.
First, we need to find the radius of the circular path of the particle for both distances.
For the first distance of 1.31 cm = 0.0131 m, the radius is r1 = 0.0131 m.
For the second distance of 2.71 cm = 0.0271 m, the radius is r2 = 0.0271 m.
Using the equation for centripetal force, we can set up the following system of equations:
k * (Q * 3.2 x 10^-19 C) / (0.0131 m)^2 = (6.70 x 10^-27 kg * (2.30 x 10^5 m/s)^2) / 0.0131 m
k * (Q * 3.2 x 10^-19 C) / (0.0271 m)^2 = (6.70 x 10^-27 kg * (3.10 x 10^6 m/s)^2) / 0.0271 m
Now, we can solve these equations simultaneously to find the value of Q, the total charge on the wire.
After finding Q, we can use the equation for linear charge density to calculate the charge density of the wire by dividing Q by the length of the wire.