How much work is required to completely separate two charges (each -1.9 microC ) and leave them at rest if they were initially 6.0mm apart?
work=INT force.dx from .006 to infinity
= int kq1q2/x^2 dx
= -kq1q1/x from .006 to inf
To calculate the work required to completely separate two charges, we can use the formula for the electric potential energy of a system of two charges:
ΔPE = k * (Q1 * Q2) / r
Where:
ΔPE is the change in potential energy
k is the Coulomb's constant (8.99 * 10^9 N m^2/C^2)
Q1 and Q2 are the magnitudes of the charges (-1.9 * 10^-6 C in this case)
r is the separation distance (6.0 mm = 6.0 * 10^-3 m)
Substituting the values into the formula, we can calculate the work required:
ΔPE = (8.99 * 10^9 N m^2/C^2) * (-1.9 * 10^-6 C) * (-1.9 * 10^-6 C) / (6.0 * 10^-3 m)
Simplifying the equation:
ΔPE = (8.99 * 10^9 N m^2/C^2) * (3.61 * 10^-12 C^2) / (6.0 * 10^-3 m)
ΔPE ≈ 5.394 * 10^-3 N m
Therefore, the work required to completely separate the two charges is approximately 5.394 * 10^-3 N m.
To find the work required to separate two charges and leave them at rest, you can use the formula:
Work = Potential Energy final - Potential Energy initial
The potential energy of two charges separated by a distance is given by:
Potential Energy = k * (Q1 * Q2) / r
Where:
- k is the Coulomb's constant (8.99 x 10^9 N m^2/C^2)
- Q1 and Q2 are the magnitudes of the charges (-1.9 μC)
- r is the initial distance between the charges (6.0 mm = 0.006 m)
Let's calculate the potential energy initial and final first:
Potential Energy initial = k * (Q1 * Q2) / r_initial
Potential Energy initial = (8.99 x 10^9 N m^2/C^2) * ((-1.9 x 10^-6 C) * (-1.9 x 10^-6 C)) / 0.006 m
Potential Energy final = k * (Q1 * Q2) / r_final
Potential Energy final = (8.99 x 10^9 N m^2/C^2) * ((-1.9 x 10^-6 C) * (-1.9 x 10^-6 C)) / ∞ [Since they are being separated to infinity, r_final → ∞]
Now, plug in the values and calculate the potential energy initial and final:
Potential Energy initial = (8.99 x 10^9 N m^2/C^2) * (3.61 x 10^-12 C^2) / 0.006 m
Potential Energy initial ≈ 18 x 10^-3 J
Potential Energy final ≈ 0 J [Since the charges are separated to infinity]
Finally, subtract the potential energy initial from the potential energy final to get the work required:
Work = Potential Energy final - Potential Energy initial
Work ≈ 0 J - 18 x 10^-3 J
Work ≈ -18 x 10^-3 J
The work required to completely separate two charges and leave them at rest is approximately -18 x 10^-3 J. The negative sign indicates that work is done by an external force against the electric force between the charges.