"Consider a long charged straight wire that lies fixed and a particle of charge +2e and mass 6.70×10-27 kg. When the particle is at a distance 1.61 cm from the wire it has a speed 2.70×105 m/s, going away from the wire. When it is at a new distance of 2.71 cm, its speed is 3.40×106 m/s. What is the charge density of the wire?"
This is my attempt.
From conservation of energy we can find the electric field:
KE(i) + U(i) = KE(f) + U(f)
1/2mv(i)2 + qV(i) = 1/2mv(f)2 + qV(f) (v here is speed, V is potential)
Some algebra leads us to this:
Δ(V) = m(v(i)2 - v(f)2 )/(2q)
Since E = Δ(V)/Δ(d)
Then:
E = m(v(i)2 - v(f)2 )/(2q * Δ(d))
I find the same answer using kinematics:
F = ma = qE and v(f)2 = v(i)2 + 2aΔd so we can find figure out E
Finally, using Gauss's law:
The field generated by an infinite (long) wire is:
E = lambda/(2piR*epsilon)
Plugging in my expression for E and working out the algebra, I end up with:
lambda = m(v(i)2 - m(f)2 )*2piR(epsilon)/(2qΔd)
Which "R" do I use here? In other words, what is the radius of my Gaussian surface? Or is my entire approach completely wrong?
Your approach is almost correct, but there is a small mistake in your calculations. Let's go through the problem and correct it.
To find the charge density of the wire, we can start with Coulomb's law, which states that the electric field created by a charged wire is given by:
E = (λ / (2πε₀R))
Where λ is the charge density (charge per unit length), ε₀ is the permittivity of vacuum, and R is the distance from the wire.
Now, let's start from the conservation of energy equation:
KE(i) + U(i) = KE(f) + U(f)
Where KE represents the kinetic energy and U represents the electric potential energy.
For the initial position, at a distance of 1.61 cm (0.0161 m) from the wire, the particle has a speed of 2.70 × 10⁵ m/s.
For the final position, at a distance of 2.71 cm (0.0271 m) from the wire, the particle has a speed of 3.40 × 10⁶ m/s.
We can now write down the equation using the given values:
(1/2)m(v(i)²) + qV(i) = (1/2)m(v(f)²) + qV(f)
Now, rearrange the equation to isolate the potential difference (V):
Δ(V) = (1/(2q))(m(v(i)² - v(f)²))
Next, we can use the relationship between electric field (E), potential difference (V), and distance (Δd):
E = Δ(V) / Δ(d)
Substituting the equation for Δ(V), we get:
E = (1/(2q))(m(v(i)² - v(f)²)) / Δ(d)
Now, we can substitute the expression for electric field (E) from Coulomb's law:
(λ / (2πε₀R)) = (1/(2q))(m(v(i)² - v(f)²)) / Δ(d)
Rearrange the equation to solve for charge density (λ):
λ = (2πε₀R(q/m))(v(i)² - v(f)²) / Δ(d)
Here, R represents the distance from the wire to the Gaussian surface where we are trying to measure the electric field. In this case, we are given the distances of 1.61 cm and 2.71 cm. It is up to you to choose the appropriate distance based on the problem and what you are trying to calculate.
So, using the corrected equation, plug in the given values for mass (m), charge (q), initial speed (v(i)), final speed (v(f)), and the chosen distance (Δ(d)), and you will be able to find the charge density (λ) of the wire.
Your approach is mostly correct, but there are a few adjustments needed.
To find the charge density of the wire, let's consider the following steps:
1. Start by finding the change in potential energy between the initial and final positions of the particle.
ΔU = q(V(final) - V(initial))
2. Use the change in potential energy to find the change in electric potential.
ΔV = ΔU / q
3. Next, calculate the change in distance (Δd) between the initial and final positions of the particle.
Δd = d(final) - d(initial)
4. Use the change in distance and the change in electric potential to calculate the electric field at the initial position.
E(initial) = ΔV / Δd
5. Apply Gauss's law to find the electric field generated by the wire at the initial position (d(initial)). The Gaussian surface chosen for an infinite straight wire is a cylinder with a radius chosen as the distance to the wire (d(initial)).
E = λ / (2πε₀d(initial))
6. Equate the value of the electric field from steps 4 and 5.
E(initial) = λ / (2πε₀d(initial))
7. Solve the equation for the charge density (λ).
λ = E(initial) * 2πε₀ * d(initial)
By substituting the given values into this final expression, you will be able to calculate the charge density of the wire.