Examine the charge distribution shown. Sphere 1 has a charge of +3.0 × 10-4
C; sphere 2 has a charge of –3.0 × 10-4
C; and sphere 3 has a charge of +3.0 × 10-4
C. Assume that Coulomb’s constant, (k) is equal to 9.0×109 Nm2C2
.
Show all your work when completing each of these questions.
Illustration of the charge distribution. Charge 1, 2 and 3 are in a straight line. Charge 1 is on the left; Charge 2 is in the middle 1.50 m away from both charge 1 and 3. Charge 3 is on the right,. A point in space is directly above the middle charge, charge 2. This point is 1.50 m away from charge 2. Both charge 1 and charge 3 are 2.12 m away from the point in space.
Determine the total electric potential energy for the charge distribution.
To determine the total electric potential energy for the charge distribution, we need to calculate the potential energy for each pair of charges and then add them together.
The potential energy between two point charges can be calculated using the formula:
U = k * (|q1 * q2|) / r
Where U is the potential energy, k is Coulomb's constant, q1 and q2 are the charges, and r is the distance between the charges.
First, let's calculate the potential energy between charge 1 and charge 2:
U_12 = (9.0 * 10^9 Nm^2/C^2) * (|3.0 * 10^-4 C| * |-3.0 * 10^-4 C|) / (1.50 m)
= (9.0 * 10^9 Nm^2/C^2) * (9.0 * 10^-8 C^2) / (1.50 m)
= 5.40 * 10^-1 J
Next, let's calculate the potential energy between charge 2 and charge 3:
U_23 = (9.0 * 10^9 Nm^2/C^2) * (|-3.0 * 10^-4 C| * |3.0 * 10^-4 C|) / (1.50 m)
= (9.0 * 10^9 Nm^2/C^2) * (9.0 * 10^-8 C^2) / (1.50 m)
= 5.40 * 10^-1 J
Finally, let's calculate the potential energy between charge 1 and the point in space:
U_1s = (9.0 * 10^9 Nm^2/C^2) * (|3.0 * 10^-4 C| * |0 C|) / (2.12 m)
= 0 J
The total potential energy for the charge distribution is the sum of these three energies:
U_total = U_12 + U_23 + U_1s
= (5.40 * 10^-1 J) + (5.40 * 10^-1 J) + (0 J)
= 1.08 J
Therefore, the total electric potential energy for the charge distribution is 1.08 Joules.
Determine the total electric potential at point Z.
To determine the total electric potential at point Z, we need to calculate the potential due to each charge and then add them together.
The electric potential due to a point charge can be calculated using the formula:
V = k * (|q|) / r
Where V is the electric potential, k is Coulomb's constant, q is the charge, and r is the distance between the charge and the point.
First, let's calculate the electric potential at point Z due to charge 1:
V_1 = (9.0 * 10^9 Nm^2/C^2) * (|3.0 * 10^-4 C|) / (2.12 m)
= (9.0 * 10^9 Nm^2/C^2) * (3.0 * 10^-4 C) / (2.12 m)
= 4.24 * 10^5 V
Next, let's calculate the electric potential at point Z due to charge 2:
V_2 = (9.0 * 10^9 Nm^2/C^2) * (|-3.0 * 10^-4 C|) / (1.50 m)
= (9.0 * 10^9 Nm^2/C^2) * (3.0 * 10^-4 C) / (1.50 m)
= 1.80 * 10^6 V
Finally, let's calculate the electric potential at point Z due to charge 3:
V_3 = (9.0 * 10^9 Nm^2/C^2) * (|3.0 * 10^-4 C|) / (2.12 m)
= (9.0 * 10^9 Nm^2/C^2) * (3.0 * 10^-4 C) / (2.12 m)
= 4.24 * 10^5 V
The total electric potential at point Z is the sum of these individual potentials:
V_total = V_1 + V_2 + V_3
= (4.24 * 10^5 V) + (1.80 * 10^6 V) + (4.24 * 10^5 V)
= 2.62 * 10^6 V
Therefore, the total electric potential at point Z is 2.62 million volts.
To determine the total electric potential energy of the charge distribution, we need to find the potential energy between the charges 1 and 2, and between charges 2 and 3, and then add them together.
First, we'll calculate the potential energy between charges 1 and 2:
The formula for the electric potential energy between two point charges is given by:
Potential Energy = k * (|Q1| * |Q2|) / r
Where:
k is the Coulomb's constant (9.0 × 10^9 Nm^2/C^2)
|Q1| and |Q2| are the magnitudes of the charges
r is the distance between the charges.
Substituting the given values:
|Q1| = |Q2| = 3.0 × 10^-4 C
r = 1.50 m
Potential Energy1-2 = (9.0 × 10^9 Nm^2/C^2) * (3.0 × 10^-4 C * 3.0 × 10^-4 C) / 1.50 m
Calculate the value to find Potential Energy1-2.
Next, we'll calculate the potential energy between charges 2 and 3:
Using the same formula, substituting the values:
|Q2| = 3.0 × 10^-4 C
r = 1.50 m
Potential Energy2-3 = (9.0 × 10^9 Nm^2/C^2) * (3.0 × 10^-4 C * 3.0 × 10^-4 C) / 1.50 m
Calculate the value to find Potential Energy2-3.
Finally, we'll add the two potential energies together to find the total electric potential energy of the charge distribution:
Total Electric Potential Energy = Potential Energy1-2 + Potential Energy2-3
Calculate the value to find the Total Electric Potential Energy.