Two point charges of +1.0 uC and -2.0 uC are located 0.50 m apart. What is the minimum amount of work needed to move the charges apart to double the distance between them? (k=8.99x10^9Nm^2/C^2)
To find the minimum amount of work needed to move the charges apart to double the distance between them, we need to calculate the initial potential energy and the final potential energy.
The potential energy, U, of two point charges is given by the formula:
U = (k * |q1 * q2|) / r
where:
- U is the potential energy
- k is the value of Coulomb's constant (8.99 x 10^9 Nm^2/C^2)
- |q1 * q2| is the absolute value of the product of the charges
- r is the separation distance between the charges
Let's calculate the initial potential energy with the initial separation distance:
U_initial = (k * |q1 * q2|) / r_initial
U_initial = (8.99 x 10^9 Nm^2/C^2 * |+1.0 x 10^-6 C * -2.0 x 10^-6 C|) / 0.50 m
U_initial = (8.99 x 10^9 Nm^2/C^2 * 2 x 10^-12 C^2) / 0.50 m
U_initial = (8.99 x 2 x 10^-3 Nm^2) / 0.50 m
U_initial = 17.98 x 10^-3 Nm
Next, we need to calculate the final separation distance. To double the distance between the charges, we multiply the initial separation distance (r_initial) by 2:
r_final = 2 * r_initial
r_final = 2 * 0.50 m
r_final = 1.00 m
Now, we can calculate the final potential energy:
U_final = (k * |q1 * q2|) / r_final
U_final = (8.99 x 10^9 Nm^2/C^2 * |+1.0 x 10^-6 C * -2.0 x 10^-6 C|) / 1.00 m
U_final = (8.99 x 10^9 Nm^2/C^2 * 2 x 10^-12 C^2) / 1.00 m
U_final = (8.99 x 2 x 10^-3 Nm^2) / 1.00 m
U_final = 17.98 x 10^-3 Nm
Finally, we can calculate the minimum amount of work needed to move the charges apart:
Work = U_final - U_initial
Work = (17.98 x 10^-3 Nm) - (17.98 x 10^-3 Nm)
Work = 0
Therefore, the minimum amount of work needed to move the charges apart to double the distance between them is equal to 0.
To find the minimum amount of work needed to move the charges apart and double the distance, we can use the formula for electric potential energy.
The formula for electric potential energy is given by:
U = (k * q1 * q2) / r
Where:
U is the electric potential energy
k is the Coulomb's constant, 8.99x10^9 Nm^2/C^2
q1 and q2 are the magnitudes of the charges
r is the separation between the charges
Let's calculate the electric potential energy for the initial configuration of the charges.
Given:
q1 = +1.0 uC = 1.0x10^-6 C
q2 = -2.0 uC = -2.0x10^-6 C
r = 0.50 m
Using the formula, we have:
U(initial) = (k * q1 * q2) / r
= (8.99x10^9 Nm^2/C^2) * (1.0x10^-6 C) * (-2.0x10^-6 C) / 0.50 m
Calculating this expression:
U(initial) = -35.96 J
Now, we need to find the final separation between the charges when the distance is doubled. The new separation will be 2 * r, which is equal to 2 * 0.50 m = 1.00 m.
Using the same formula, let's calculate the final electric potential energy.
U(final) = (k * q1 * q2) / (2 * r)
= (8.99x10^9 Nm^2/C^2) * (1.0x10^-6 C) * (-2.0x10^-6 C) / (2 * 0.50 m)
Calculating this expression:
U(final) = -17.98 J
Finally, we can find the minimum amount of work needed to move the charges apart by taking the difference between U(final) and U(initial):
Work = U(final) - U(initial)
= (-17.98 J) - (-35.96 J)
= 17.98 J + 35.96 J
= 53.94 J
Therefore, the minimum amount of work needed to move the charges apart and double the distance between them is 53.94 J.