Two point charges, 4.0 µC and -2.0 µC, are placed 5.4 cm apart on the x axis, such that the -2.0 µC charge is at x = 0 and the 4.0 µC charge is at x = 5.4 cm.

(a) At what point(s) along the x axis is the electric field zero? (If there is no point where E = 0 in a region, enter "0" in that box.)
x < 0 cm
(b) At what point(s) along the x axis is the potential zero? Let V = 0 at r = . (If there is no point where V = 0 in a region, enter "0" in that box.)
x < 0 cm
0 < x < 5.4 cm cm

To determine the points along the x-axis where the electric field and potential are zero, we need to use the equations for electric field and electric potential due to point charges.

(a) Electric Field:

The formula for the electric field (E) due to a point charge is given by:

E = k * q / r^2

where E is the electric field, k is the electrostatic constant (9 x 10^9 Nm^2/C^2), q is the charge, and r is the distance from the charge.

Let's consider the electric field due to the 4.0 µC charge at a point x cm away:

E1 = (9 x 10^9 Nm^2/C^2) * (4.0 x 10^-6 C) / (x cm)^2

Similarly, the electric field due to the -2.0 µC charge at a point (5.4 - x) cm away is:

E2 = (9 x 10^9 Nm^2/C^2) * (-2.0 x 10^-6 C) / ((5.4 - x) cm)^2

The total electric field at any point is the sum of these two fields:

E_total = E1 + E2

We need to find the value(s) of x for which the electric field is zero. To do this, set E_total = 0 and solve for x.

E1 + E2 = 0

(9 x 10^9 Nm^2/C^2) * (4.0 x 10^-6 C) / (x cm)^2 + (9 x 10^9 Nm^2/C^2) * (-2.0 x 10^-6 C) / ((5.4 - x) cm)^2 = 0

Solve this equation to find the value(s) of x where the electric field is zero.

(b) Electric Potential:

The electric potential (V) due to a point charge is given by:

V = k * q / r

where V is the potential, k is the electrostatic constant (9 x 10^9 Nm^2/C^2), q is the charge, and r is the distance from the charge.

Similar to the electric field, we can calculate the potential due to each charge and then find the total potential at any given point on the x-axis.

V1 = (9 x 10^9 Nm^2/C^2) * (4.0 x 10^-6 C) / (x cm)

V2 = (9 x 10^9 Nm^2/C^2) * (-2.0 x 10^-6 C) / (5.4 - x cm)

The total potential at any point is the sum of these two potentials:

V_total = V1 + V2

Set V_total = 0 and solve for x to find the value(s) of x where the potential is zero.

Solve this equation to find the value(s) of x where the potential is zero.