A +3.0 nC charge is at x=0 cm and a -1.0 nC charge is at x=4 cm. At what point or points on the x-axis is the electric potential zero?

Well, looks like these charges have some "potential" for creating a zero electric potential point. Let's find out where that magical spot is on the x-axis!

First, we need to find the distance between the charges. In this case, it's 4 cm - 0 cm, which gives us a distance of 4 cm.

To find the point(s) where the electric potential is zero, we need to consider the formula for electric potential due to a point charge, which is V = k * q / r, where V is the electric potential, k is the Coulomb's constant, q is the charge, and r is the distance.

Now, we'll plug in the values. Let's say the point where the potential is zero is at x cm from the positive charge. So, the distance from the negative charge would then be 4 - x cm.

Setting up the equation for the electric potential, we have:

(k * 3.0 nC) / x = (k * -1.0 nC) / (4 - x)

Now, let's multiply both sides by x and (4 - x) to get rid of the denominators:

3.0 nC * (4 - x) = -1.0 nC * x

Simplifying, we get:

12 - 3x = -x

Rearranging, we find:

2x = 12

Solving for x, we get x = 6 cm.

So, the electric potential is zero at the point x = 6 cm on the x-axis.

But wait, there's more! Since the problem asks for "point(s)" where the electric potential is zero, let's also consider the case where x = 0 cm. If we plug in this value into the equation, we see that the electric potential is also zero at the point x = 0 cm.

So, to sum it up, the electric potential is zero at two points on the x-axis: x = 0 cm and x = 6 cm.

To find the point or points on the x-axis where the electric potential is zero, we can use the formula for electric potential due to a point charge:

V = k * q / r

Where:
V is the electric potential,
k is the Coulomb's constant (8.99 x 10^9 Nm^2/C^2),
q is the charge, and
r is the distance from the charge.

Let's calculate the electric potential at two points on the x-axis, one on each side of the origin (x=0).

1. Point A at x = -d cm:

The electric potential due to the +3.0 nC charge at point A can be calculated as follows:

V_A = k * q_A / r_A

Where:
k = 8.99 x 10^9 Nm^2/C^2 (Coulomb's constant),
q_A = +3.0 nC (+3.0 x 10^-9 C) (charge at x=0),
r_A = -d cm (-ve sign because point A is on the left side of the origin).

2. Point B at x = d cm:

The electric potential due to the -1.0 nC charge at point B can be calculated as follows:

V_B = k * q_B / r_B

Where:
k = 8.99 x 10^9 Nm^2/C^2 (Coulomb's constant),
q_B = -1.0 nC (-1.0 x 10^-9 C) (charge at x=4 cm),
r_B = d cm (distance from the charge is d cm since point B is on the right side of the origin).

For the electric potential to be zero, V_A + V_B must equal zero.

So, we can set up the equation as follows:

k * q_A / r_A + k * q_B / r_B = 0

Plugging in the known values, we have:

(8.99 x 10^9 Nm^2/C^2 * 3.0 x 10^-9 C) / (-d cm) + (8.99 x 10^9 Nm^2/C^2 * -1.0 x 10^-9 C) / d cm = 0

Simplifying the equation further:

(8.99 x 3.0 - 8.99 x 1.0) / (d cm * d cm) = 0

(26.97 - 8.99) / (d cm * d cm) = 0

17.98 / (d cm * d cm) = 0

Since the numerator is non-zero, in order for the equation to be satisfied, the denominator (d cm * d cm) must be infinity.

Therefore, the electric potential will be zero at a point infinitely far from the charges on the x-axis.

To determine the point or points on the x-axis where the electric potential is zero, we need to consider the electric potential due to both charges.

The electric potential at a point due to a point charge can be calculated using the formula:

V = k * (q / r),

where V is the electric potential, k is the electrostatic constant (k = 8.99 x 10^9 Nm^2/C^2), q is the charge, and r is the distance from the charge to the point where the electric potential is being calculated.

Let's denote the distance from the +3.0 nC charge to the point as x1, and the distance from the -1.0 nC charge to the point as x2.

For the electric potential to be zero, the sum of the electric potentials due to both charges must equal zero. Mathematically, this can be expressed as:

k * (q1 / x1) + k * (q2 / x2) = 0,

where q1 is the charge of the +3.0 nC charge, and q2 is the charge of the -1.0 nC charge.

Using the given values, we have:

(8.99 x 10^9 Nm^2/C^2) * (3.0 x 10^-9 C / x1) + (8.99 x 10^9 Nm^2/C^2) * (-1.0 x 10^-9 C / x2) = 0.

Simplifying the equation further, we get:

(8.99 / x1) - (8.99 / x2) = 0.

To find the values of x1 and x2 that satisfy this equation, we can re-arrange the equation as:

1 / x1 = 1 / x2.

Taking the reciprocal of both sides, the equation becomes:

x1 = x2.

Therefore, the electric potential will be zero at the point where the distances from both charges are equal. In this case, the electric potential will be zero at the point on the x-axis where the distances from the +3.0 nC charge and the -1.0 nC charge are both the same.

Wherever 3/|x| - 1/|4-x| = 0 , the electric potential is zero.

|x| is the distance from the +3 nC charge and |4-x| is the distance from the -1 nC charge.

To omit the absolute value | | signs and solve algebraically, you need to write and solve separate equations for x<0, 0<x<4 and x>4

For 0<x<4,
3/x - 1/(4-x) = 0
3/x = 1/(4-x)
12 - 3x = x
x = 3

See if there are solutions in the other regions.