Three point charges of 2.0 μ C, 4.0 μ C, and 6.0 μ C form an equilateral triangle with each side 2.0cm.
Find the electric potential energy of this distribution.
To find the electric potential energy of this distribution, we can use the formula:
U = k * (q1 * q2 / r)
Where U is the electric potential energy, k is the electrostatic constant (k = 9 * 10^9 Nm^2/C^2), q1 and q2 are the magnitudes of the charges, and r is the distance between the charges.
Given that we have an equilateral triangle with each side measuring 2.0 cm, we can calculate the distance between the charges using the Pythagorean theorem:
r = √(h^2 + (2/3 * a)^2)
where h is the height of the equilateral triangle and a is the length of each side.
First, let's find the height (h) of the equilateral triangle:
h = √(a^2 - (a/2)^2)
= √(4 - 1)
= √3 cm
Now we can calculate the distance (r) between the charges:
r = √(h^2 + (2/3 * a)^2)
= √(3 + (4/3)^2)
= √(3 + 16/9)
= √(27/9 + 16/9)
= √(43/9) cm
Given that q1 = 2.0 μC, q2 = 4.0 μC, and q3 = 6.0 μC, we can calculate the electric potential energy (U).
U1 = k * (q1 * q2 / r)
= (9 * 10^9 Nm^2/C^2) * ((2.0 * 10^-6 C) * (4.0 * 10^-6 C) / (√(43/9) * 10^-2 m))
= 7.43 * 10^-5 Joules
U2 = k * (q2 * q3 / r)
= (9 * 10^9 Nm^2/C^2) * ((4.0 * 10^-6 C) * (6.0 * 10^-6 C) / (√(43/9) * 10^-2 m))
= 1.86 * 10^-4 Joules
U3 = k * (q1 * q3 / r)
= (9 * 10^9 Nm^2/C^2) * ((2.0 * 10^-6 C) * (6.0 * 10^-6 C) / (√(43/9) * 10^-2 m))
= 7.43 * 10^-5 Joules
Now, we can find the total electric potential energy (U) of the distribution:
U = U1 + U2 + U3
= 7.43 * 10^-5 Joules + 1.86 * 10^-4 Joules + 7.43 * 10^-5 Joules
= 3.01 * 10^-4 Joules
Therefore, the electric potential energy of this distribution is 3.01 * 10^-4 Joules.
To find the electric potential energy of this distribution, we can use the formula:
ΔPE = K * q1 * q2 / r
where ΔPE is the change in electric potential energy, K is the electromagnetic constant (8.99 × 10^9 N m^2/C^2), q1 and q2 are the magnitudes of the charges, and r is the distance between the charges.
In this case, we have an equilateral triangle, so all sides have a length of 2.0 cm. To find the distance between the charges, we can use the formula for the height of an equilateral triangle:
h = (√3/2) * s
where h is the height and s is the length of each side.
Let's calculate the height of the equilateral triangle:
h = (√3/2) * 2.0 cm
h = √3 cm
Now, we can calculate the electric potential energy:
ΔPE = K * q1 * q2 / r
ΔPE = (8.99 × 10^9 N m^2/C^2) * (2.0 μC) * (4.0 μC) / (√3 cm)
Note that we need to convert the charges from microcoulombs (μC) to coulombs (C) to match the units of the electromagnetic constant.
1 μC = 1 × 10^-6 C
ΔPE = (8.99 × 10^9 N m^2/C^2) * (2.0 × 10^-6 C) * (4.0 × 10^-6 C) / (√3 cm)
Simplifying the calculation:
ΔPE ≈ (8.99 × 2 × 4) / (√3) * 10^3 N m
Finally, we can evaluate this expression to find the electric potential energy of the distribution.