The figure below shows two pint charges, each of -2Q, fixed on the y-axis at y = +a and at y = -a . A third

points charge -Q is placed on the x-axis at x = 2a . Express all algebraic answers in terms of Q, a, and
fundamental constants.
(a) What is the net force (magnitude and direction) on a charge -Q due to the other two charges? (7 points)
(b) What is the net electric field at the origin due to all three charges? (7 points)
(c) What is the electrical potential at the origin due to all three charges? (6 points)

Pint charges or point charges?

What figure below?

(a) The net force on a charge -Q due to the other two charges can be found using Coulomb's Law.

The force between the charge -Q and one of the charges -2Q can be calculated as:
F1 = k * |-Q| * |-2Q| / (2a)^2
= 2kQ^2 / (4a^2)
= kQ^2 / (2a^2)

Since the charges are of the same magnitude, the net force will be twice the force F1:
F_net = 2F1
= 2 * (kQ^2 / (2a^2))
= kQ^2 / a^2

The direction of the net force will be along the negative x-axis since the charge -Q is repelled by the two negative charges on the y-axis.

Therefore, the net force on the charge -Q due to the other two charges is kQ^2 / a^2, directed along the negative x-axis.

(b) The net electric field at the origin due to all three charges can be found by summing the electric fields from each individual charge.

The electric field from one of the charges -2Q can be calculated as:
E1 = k * |-2Q| / (2a)^2
= 2kQ / (4a^2)
= kQ / (2a^2)

Since the charges are symmetric and have the same magnitude, the horizontal components of the electric fields from both charges on the y-axis cancel out. Therefore, only the vertical components contribute to the net electric field at the origin.

The vertical component of the electric field from each charge on the y-axis has the same magnitude but opposite direction. Therefore, the net electric field at the origin due to the charges on the y-axis will be zero.

The electric field from the charge -Q on the x-axis at the origin can be calculated as:
E2 = k * |-Q| / (2a)^2
= kQ / (4a^2)

The net electric field at the origin is the vector sum of E1 and E2, which are perpendicular to each other.

Therefore, the net electric field at the origin due to all three charges is:
E_net = sqrt((E1)^2 + (E2)^2)
= sqrt((kQ / (2a^2))^2 + (kQ / (4a^2))^2)
= sqrt((k^2Q^2 + k^2Q^2/4) / a^4)
= sqrt((5k^2Q^2 / 4) / a^4)
= sqrt(5k^2Q^2) / (2a^2)
= (sqrt(5)kQ) / (2a^2)

(c) The electrical potential at the origin due to all three charges can be found by summing the potentials from each individual charge.

The potential due to one of the charges -2Q can be calculated as:
V1 = k * |-2Q| / (2a)
= 2kQ / (2a)
= kQ / a

Since the charges on the y-axis are symmetric and have the same magnitude, their potentials will cancel out at the origin.

The potential due to the charge -Q on the x-axis at the origin can be calculated as:
V2 = k * |-Q| / (2a)
= kQ / (2a)

The net potential at the origin is the sum of V1 and V2.

Therefore, the electrical potential at the origin due to all three charges is:
V_net = V1 + V2
= (kQ / a) + (kQ / (2a))
= (2kQ + kQ) / (2a)
= (3kQ) / (2a)

To find the net force on a charge -Q due to the other two charges, we can use Coulomb's law. Coulomb's law states that the force between two charges is given by the equation:

F = k * (|q1| * |q2|) / r^2

Where F is the force between the charges, k is the electrostatic constant (approximately equal to 8.99 × 10^9 N m^2/C^2), q1 and q2 are the magnitudes of the charges, and r is the distance between them.

Let's start with part (a):

(a) To find the net force on a charge -Q due to the other two charges, we need to calculate the forces between -Q and the two -2Q charges separately, and then find their vector sum.

The force between -Q and the -2Q charge fixed at y = +a can be calculated as follows:

F1 = k * (|-Q| * |2Q|) / (2a)^2

The direction of this force will be along the positive x-axis.

Now, the force between -Q and the -2Q charge fixed at y = -a can be calculated as:

F2 = k * (|-Q| * |2Q|) / (4a)^2

The direction of this force will also be along the positive x-axis.

To find the net force, we need to add these two forces together:

Net Force = F1 + F2

Next, let's move on to part (b):

(b) To find the net electric field at the origin due to all three charges, we can use the concept of superposition. The electric field at a point is the vector sum of the electric fields created by each charge.

The electric field due to a point charge is given by Coulomb's law as:

E = k * (|q|) / r^2

To find the electric field at the origin, we need to calculate the electric fields due to each charge separately and then find their vector sum.

The electric field at the origin due to the -2Q charges at y = +a and y = -a can be calculated as follows:

E1 = k * (|-2Q|) / a^2

E2 = k * (|-2Q|) / (3a)^2

The direction of these electric fields will be along the positive y-axis and negative y-axis, respectively.

Now, the electric field at the origin due to the -Q charge at x = 2a can be calculated as:

E3 = k * (|-Q|) / (2a)^2

The direction of this electric field will be along the negative x-axis.

To find the net electric field at the origin, we need to add these three electric fields together:

Net Electric Field = E1 + E2 + E3

Finally, let's move on to part (c):

(c) To find the electrical potential at the origin due to all three charges, we can use the concept of potential energy. The electrical potential at a point is the sum of the potential energies created by each charge.

The potential energy due to a point charge is given by the equation:

V = k * (|q|) / r

To find the electrical potential at the origin, we need to calculate the potential energies due to each charge separately and then sum them up.

The potential energy at the origin due to the -2Q charges at y = +a and y = -a can be calculated as follows:

V1 = k * (|-2Q|) / a

V2 = k * (|-2Q|) / (3a)

Now, the potential energy at the origin due to the -Q charge at x = 2a can be calculated as:

V3 = k * (|-Q|) / (2a)

To find the electrical potential at the origin, we need to add these three potential energies together:

Electrical Potential = V1 + V2 + V3

Remember to express all final answers in terms of Q, a, and fundamental constants like k.