two -8.1 nC charges, are each separated by 5.0 mm from a proton. If the two negative charges are held fixed at their locations and the proton is given an initial velocity v, what is the minimum initial speed v that the proton needs to totally escape from the negative charges? (k=1/4πε 0 = 8.99 × 109 N · m2/C2 , e = 1.60 × 10-19 C, m proton = 1.67 x 10-27 kg)
Well, this sounds like a shocking situation! Let's electrify our brains and solve it step by step.
First, let's calculate the electric force between each of the negative charges (-8.1 nC) and the proton. We can use Coulomb's law for that. Considering the distance of 5.0 mm between them:
F = k * (q1 * q2) / r^2
F = (8.99 × 10^9 N · m^2/C^2) * ((-8.1 × 10^-9 C) * (1.6 × 10^-19 C)) / (0.005 m)^2
Alright, let me crunch those numbers. *calculating noises*
(ten minutes later...)
Okay, so the electric force between each negative charge and the proton is approximately -4.5897 x 10^-13 N.
Now, since we have 2 negative charges, we need to double that force:
F_total = 2 * (-4.5897 x 10^-13 N)
Great! Now, in order for the proton to escape from the negative charges, its initial kinetic energy must be equal to or greater than the work done against the electric force on it.
The work done against the electric force is given by:
W = F_total * d (where d is the displacement)
Since the proton needs to escape, the force is acting in the opposite direction of the displacement, so the work done is negative.
Now, let's say the proton needs to travel a distance x before escaping. The work done would be:
W = -F_total * x
We can also express the work done in terms of kinetic energy:
W = ΔKE
Where ΔKE is the change in kinetic energy. And since the proton starts from rest, we have:
ΔKE = KE_final - KE_initial
Since KE_initial = 0 (starting from rest), we can simplify to:
W = KE_final
Substituting the values, we have:
KE_final = -F_total * x
Now, using the kinetic energy equation:
KE = (1/2) * m * v^2 (where m is the mass of the proton and v is the velocity)
We can solve for the minimum speed required for the proton to escape:
KE_final = (1/2) * m * v^2
- F_total * x = (1/2) * m * v^2
Finally, rearranging the equation to solve for v:
v^2 = - 2 * (F_total * x) / m
Let me plug in the numbers and calculate the minimum speed required for the proton to escape from the negative charges. *calculating noises*
(ten minutes later...)
Oops, it seems we have encountered a complex number in the process. It seems like my electrifying humor isn't always compatible with complex physics problems. I apologize for not being able to provide you with an answer this time. Let's hope the next question is a bit lighter!
To find the minimum initial speed (v) that the proton needs to totally escape from the negative charges, we need to calculate the electrical potential energy between the proton and the negative charges.
The electrical potential energy (U) between two charges can be calculated using the equation:
U = k * (q1 * q2) / r
Where:
- U is the electrical potential energy
- k is the electrostatic constant (k = 1 / (4πε₀), where ε₀ is the permittivity of free space)
- q1 and q2 are the charges
- r is the distance between the charges
In this case, the electrical potential energy between the proton and one of the negative charges can be calculated as:
U₁ = k * (q₁ * qₚ) / r₁
Similarly, the electrical potential energy between the proton and the other negative charge can be calculated as:
U₂ = k * (q₂ * qₚ) / r₂
Since the charges are held fixed at their locations, the total initial electrical potential energy (U_total) can be found by adding the potential energies together:
U_total = U₁ + U₂
To escape from the negative charges, the proton's kinetic energy must be equal to or greater than the total electrical potential energy. The kinetic energy (K) of the proton is given by:
K = (1/2) * mₚ * v^2
Where:
- K is the kinetic energy
- mₚ is the mass of the proton
- v is the velocity of the proton
For the proton to totally escape, the kinetic energy should be equal to or greater than the total electrical potential energy.
Setting K equal to U_total:
(1/2) * mₚ * v^2 = U_total
Now we can substitute the known values into the equation and solve for v.
First, calculate the potential energies:
U₁ = (8.99 × 10^9 N · m²/C²) * ((-8.1 × 10^(-9) C) * (1.60 × 10^(-19) C)) / 0.005 m
U₂ = (8.99 × 10^9 N · m²/C²) * ((-8.1 × 10^(-9) C) * (1.60 × 10^(-19) C)) / 0.005 m
Next, calculate the total electrical potential energy:
U_total = U₁ + U₂
Now, substitute the values into the equation:
(1/2) * (1.67 × 10^(-27) kg) * v^2 = U_total
Solve for v:
v = √((2 * U_total) / (mₚ))
Finally, substitute the calculated value of U_total and the mass of the proton into the equation to find the minimum initial speed (v) needed for the proton to escape.