The drawing shows a square, each side of which has a length of L = 0.25 m. Two different positive charges q1 and q2 are fixed at the corners of the square. Find the electric potential energy of a third charge q3 = -7.00 10-9 C placed at corner A and then at corner B.
diagonal of the square = .25 * √(2)
√(2) is the square root of 2 (which equals 1.414)
corner A:
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k*q1/0.35 =(9*10^9)*(1.5*10^-9)/0.35 =38.6 V
k*q2/0.25 =(9*10^9)*(4*10^-9)/0.25 =144 V
then we add these to get the sum
38.6+144 = 182.6 V.
PE = q3 * (sum of V) = (-7.00*10^-9)* 182.6 = -1.2782*10^-6 J
Corner B:
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k*q1/0.25 =(9*10^9)*(1.5*10^-9)/0.25 =54 V
k*q2/0.35 =(9*10^9)*(4*10^-9)/0.35 =102.9 V
then we add these to get the sum
54+102.9 = 156.9 V.
PE = q3 * (sum of V) = (-7.00*10^-9)* 156.9 = -1.0983*10^-6 J
Well, well, well, looks like this charge party just got a little more interesting! So we have some charges having a square dance, huh? Let's see what's going on.
First, let's calculate the electric potential energy at corner A with charge q3. So q1 and q3 are gonna tango together, while q2 is watching from the sidelines. The equation for electric potential energy (U) is given by U = k * (q1 * q3) / r1, where k is the electrostatic constant and r1 is the distance between q1 and q3.
Now, since we have a square, the distance between q1 and q3 is simply the length of one of the sides of the square. So r1 = L = 0.25 m.
Plug in the values, do some fancy mathy moves, and you'll have the electric potential energy at corner A.
Now, let's switch things up and move q3 over to corner B. Time for a new partner, q2!
Repeat the same steps as before, using the distance between q2 and q3 (which is also L = 0.25 m) instead of r1. This time, you'll get the electric potential energy at corner B.
And that's how you calculate the electric potential energy of charge q3 at both corner A and corner B. I hope this electrifying answer charged you up with some knowledge!
To find the electric potential energy of charge q3 at corner A and then at corner B, we need to use the formula for electric potential energy:
U = k * |q1 * q3| / r1 + k * |q2 * q3| / r2
where U is the electric potential energy, k is the Coulomb's constant (9.0 x 10^9 Nm^2/C^2), q1 and q2 are the charges at the corners of the square, q3 is the charge being moved, r1 is the distance from q1 to q3, and r2 is the distance from q2 to q3.
Given:
q1 = q2 = L = 0.25 m
q3 = -7.00 x 10^-9 C
To find U at corner A:
r1 = diagonal of the square = L * sqrt(2)
r2 = L
Substituting the values into the formula:
U_A = k * |q1 * q3| / r1 + k * |q2 * q3| / r2
= (9.0 x 10^9 Nm^2/C^2) * |(0.25 m) * (-7.00 x 10^-9 C)| / (L * sqrt(2)) + (9.0 x 10^9 Nm^2/C^2) * |(0.25 m) * (-7.00 x 10^-9 C)| / (0.25 m)
= (9.0 x 10^9 Nm^2/C^2) * (0.25 m) * (-7.00 x 10^-9 C) / (L * sqrt(2)) + (9.0 x 10^9 Nm^2/C^2) * (0.25 m) * (-7.00 x 10^-9 C) / (0.25 m)
Simplifying the expression:
U_A = (9.0 x 10^9 Nm^2/C^2) * (0.25 m) * (-7.00 x 10^-9 C) / (L * sqrt(2)) + (9.0 x 10^9 Nm^2/C^2) * (0.25 m) * (-7.00 x 10^-9 C) / (0.25 m)
= - (9.0 x 10^9 Nm^2/C^2) * (0.25 m) * (7.00 x 10^-9 C) / (L * sqrt(2)) - (9.0 x 10^9 Nm^2/C^2) * (0.25 m) * (7.00 x 10^-9 C)
Similarly, for corner B:
r1 = L
r2 = diagonal of the square = L * sqrt(2)
Substituting the values into the formula:
U_B = k * |q1 * q3| / r1 + k * |q2 * q3| / r2
= (9.0 x 10^9 Nm^2/C^2) * (0.25 m) * (-7.00 x 10^-9 C) / (0.25 m) + (9.0 x 10^9 Nm^2/C^2) * (0.25 m) * (-7.00 x 10^-9 C) / (L * sqrt(2))
= (9.0 x 10^9 Nm^2/C^2) * (0.25 m) * (-7.00 x 10^-9 C) / (L * sqrt(2)) + (9.0 x 10^9 Nm^2/C^2) * (0.25 m) * (-7.00 x 10^-9 C) / (0.25 m)
Simplifying the expression:
U_B = (9.0 x 10^9 Nm^2/C^2) * (0.25 m) * (-7.00 x 10^-9 C) / (L * sqrt(2)) + (9.0 x 10^9 Nm^2/C^2) * (0.25 m) * (-7.00 x 10^-9 C)
= - (9.0 x 10^9 Nm^2/C^2) * (0.25 m) * (7.00 x 10^-9 C) / (L * sqrt(2)) - (9.0 x 10^9 Nm^2/C^2) * (0.25 m) * (7.00 x 10^-9 C)
Therefore, the electric potential energy at corner A is - (9.0 x 10^9 Nm^2/C^2) * (0.25 m) * (7.00 x 10^-9 C) / (L * sqrt(2)) - (9.0 x 10^9 Nm^2/C^2) * (0.25 m) * (7.00 x 10^-9 C), and the electric potential energy at corner B is also - (9.0 x 10^9 Nm^2/C^2) * (0.25 m) * (7.00 x 10^-9 C) / (L * sqrt(2)) - (9.0 x 10^9 Nm^2/C^2) * (0.25 m) * (7.00 x 10^-9 C).
To find the electric potential energy of the third charge q3, we need to use the formula for electric potential energy:
U = k(q1)(q3)/d1 + k(q2)(q3)/d2
Where:
- U is the electric potential energy
- k is the electrostatic constant (k = 9 x 10^9 Nm²/C²)
- q1 and q2 are the respective charges at the corners of the square
- q3 is the charge placed at corner A and then at corner B
- d1 and d2 are the distances between q1 and q3, and q2 and q3, respectively.
First, let's calculate the distances:
- For corner A, the distance from q1 is the length of the square's side L = 0.25 m.
- For corner A, the distance from q2 is the diagonal of the square, which can be calculated using Pythagoras' theorem:
d1 = √(L² + L²) = √(2L²) = √(2 * (0.25)²) = √0.125 = 0.35 m
Now, let's calculate the electric potential energy when q3 is placed at corner A:
U1 = k(q1)(q3)/d1 + k(q2)(q3)/d2
= (9 x 10^9 Nm²/C²)(q1)(-7.00 x 10^-9 C)/(0.25 m) + (9 x 10^9 Nm²/C²)(q2)(-7.00 x 10^-9 C)/(0.35 m)
Similarly, we can calculate the distance for corner B:
- For corner B, the distance from q1 is also the length of the square's side L = 0.25 m.
- For corner B, the distance from q2 is also the diagonal of the square d2, which is the same as above: d2 = 0.35 m.
Now, let's calculate the electric potential energy when q3 is placed at corner B:
U2 = k(q1)(q3)/d1 + k(q2)(q3)/d2
= (9 x 10^9 Nm²/C²)(q1)(-7.00 x 10^-9 C)/(0.25 m) + (9 x 10^9 Nm²/C²)(q2)(-7.00 x 10^-9 C)/(0.35 m)
Compute U1 and U2 to find the electric potential energy of q3 when placed at corner A and corner B, respectively.