A point charge with charge q1 = 3.10μC is held stationary at the origin. A second point charge with charge q2 = -4.60μC moves from the point ( 0.160m , 0) to the point ( 0.270m , 0.255m ). How much work W is done by the electric force on the moving point charge?
Express your answer in joules. Use k = 8.99×109N⋅m^2/C^2 for Coulomb's constant: k=1/4πϵ0.
W=_____ J
To calculate the work done by the electric force on the moving point charge, we need to use the formula for work, which is given by:
W = q2 * V
where W is the work done, q2 is the charge of the second point charge, and V is the electric potential difference.
To find the electric potential difference, we need to calculate the electric potential at the two given points and then subtract the electric potentials.
The electric potential at a point due to a single point charge is given by the formula:
V = k * q1 / r
where V is the electric potential, k is Coulomb's constant, q1 is the charge of the point charge at the origin, and r is the distance between the point charge and the point where we want to calculate the electric potential.
Let's calculate the electric potentials at the two given points:
For the point (0.160m, 0):
r1 = 0.160m (distance to the origin)
V1 = (8.99x10^9 N⋅m^2/C^2) * (3.10x10^-6 C) / (0.160m)
For the point (0.270m, 0.255m):
r2 = √(0.270^2 + 0.255^2) (distance to the origin)
V2 = (8.99x10^9 N⋅m^2/C^2) * (3.10x10^-6 C) / (√(0.270^2 + 0.255^2))
Now, subtract the electric potentials to find the electric potential difference:
ΔV = V2 - V1
After calculating ΔV, we can substitute it back into the formula for work to find the value of W.
Finally, express the answer in joules.
W = q2 * ΔV
Substitute the values of q2 and ΔV into the equation to find W.
To calculate the work done by the electric force on the moving point charge, we need to use the formula:
W = q2 * (ΔV)
where W is the work done, q2 is the charge of the moving point charge, and ΔV is the change in electric potential.
Step 1: Calculate the distance between the two points
To find the distance between the two points (0.160m, 0) and (0.270m, 0.255m), we use the Pythagorean theorem:
d = √((Δx)^2 + (Δy)^2)
where d is the distance, Δx = 0.270m - 0.160m = 0.11m, and Δy = 0.255m.
Substituting the values:
d = √((0.11m)^2 + (0.255m)^2)
d ≈ √(0.0121m^2 + 0.065025m^2)
d ≈ √0.077125m^2
d ≈ 0.277758m
Step 2: Calculate the electric potential difference ΔV
The electric potential difference ΔV between the two points can be calculated using the formula:
ΔV = (k * q1 * q2) / d
where k is Coulomb's constant, q1 and q2 are the charges of the two point charges, and d is the distance between them.
Substituting the values:
ΔV = (8.99x10^9 N·m^2/C^2) * (3.10x10^-6 C) * (-4.60x10^-6 C) / 0.277758m
ΔV ≈ -1.47069x10^3 J/C
Step 3: Calculate the work done W
Finally, we can substitute the values into the work formula to calculate the work done:
W = q2 * ΔV
W = (-4.60x10^-6 C) * (-1.47069x10^3 J/C)
W ≈ 6.764314x10^-3 J
Therefore, the work done by the electric force on the moving point charge is approximately 6.764314x10^-3 Joules.