Two point charges, 3.0 µC and -2.0 µC, are placed 4.8 cm apart on the x axis, such that the -2.0 µC charge is at x = 0 and the 3.0 µC charge is at x = 4.8 cm. At what point(s) along the x axis is the potential zero?
I know that between the two charges, the answer is 1.92 cm by solving this equation -2.0µC(x-0.048m)^2 = 3.0µC(x^2) for x, and I know that when x > 4.8, there is no point where the potential is zero. I found out that when x < 0, the answer is -9.6, but I don't know how to arrive at that answer. If anyone can show me the process to getting that answer, I'd be very appreciative.
Never mind. I figured it out.
I asks where the potential is zero, not necessarily where the force is zero
You can add the potential due to the two charges on the x axis
V = k Q/R1 = k Q1 /|x-xq|
forget k and the 10^-6 because they are the same for both and we are looking for zero
I will do the - 2 first
-2/|x|
then the +3 charge
+3/|x-4.8|
where will the sum be 0 ?
look at negative x first
d is distance left of x = 0
-2/d +3/(4.8+d)=0
3 d = 9.6 + 2 d
d = 9.6 left of zero
x = -9.6 (The only place)
look between x = 0 and x = 4.8
d is now right of x = 0
-2/d + 3/(4.8-d) = 0
3 d =+9.6 -2d
d = 9.6 nope, x not between the two
look right of x = 4.8, d is x value must be >4.8
-2/d + 3/(d-4.8) = 0
3 d = -2 d + 9.6
5 d = 9.6
d = 1.92 Nope, not to right of 4.8
The only place where there can be no potential is on the opposite side of the 2 charge from the 3 charge, negative x
To find the point(s) along the x-axis where the potential is zero, we need to set up the equation for the potential due to each charge and solve for x.
Let's denote the distance from the point on the x-axis where the potential is zero as "x0".
The potential due to the positive charge of 3.0 µC can be calculated using the equation:
V1 = k * q1 / r1
where:
V1 is the potential due to the positive charge,
k is the Coulomb's constant (9.0 x 10^9 Nm^2/C^2),
q1 is the charge (3.0 µC), and
r1 is the distance from the positive charge to the point (x0).
Similarly, the potential due to the negative charge of -2.0 µC can be calculated using the equation:
V2 = k * q2 / r2
where:
V2 is the potential due to the negative charge,
q2 is the charge (-2.0 µC), and
r2 is the distance from the negative charge to the point (x0).
Since we want to find the value of x0 at which the potential is zero, we can set up the equation:
V1 + V2 = 0
Substituting the formulas for V1 and V2:
(k * q1 / r1) + (k * q2 / r2) = 0
Plugging in the values: q1 = 3.0 µC, q2 = -2.0 µC, and r2 = 4.8 cm = 0.048 m:
(9.0 x 10^9 Nm^2/C^2 * 3.0 x 10^-6 C / r1) - (9.0 x 10^9 Nm^2/C^2 * 2.0 x 10^-6 C / 0.048 m) = 0
Now, rearranging the equation and solving for r1:
(9.0 x 10^9 Nm^2/C^2 * 3.0 x 10^-6 C) / r1 = (9.0 x 10^9 Nm^2/C^2 * 2.0 x 10^-6 C) / 0.048 m
Canceling out the units and solving for r1:
r1 = [(9.0 x 10^9 Nm^2/C^2 * 2.0 x 10^-6 C) * 0.048 m] / (9.0 x 10^9 Nm^2/C^2 * 3.0 x 10^-6 C)
Simplifying the expression:
r1 = 0.096 m
Therefore, the potential is zero at a point 0.096 m (or 9.6 cm) to the left of the negative charge.
To find the point(s) along the x-axis where the potential is zero, you need to consider the electric potential created by each charge separately.
Let's start with the case when x < 0: In this scenario, the point of interest is to the left of the -2.0 µC charge. The potential due to a point charge is given by the equation V = k * (q / r), where V is the electric potential, k is Coulomb's constant (approximately 9 × 10^9 Nm^2/C^2), q is the charge, and r is the distance from the charge to the point of interest.
For the -2.0 µC charge at x = 0, the potential at any point to the left (x < 0) is given by V1 = k * (-2.0 µC) / |x - 0|. Substituting the values, we have V1 = k * 2.0 × 10^(-6) C / |x|.
Since we want the potential to be zero, we can set V1 = 0 and solve for x:
0 = k * 2.0 × 10^(-6) C / |x|.
Using the equation above, we find that |x| = ∞ (since dividing any number by zero gives infinity).
However, negative infinity (-∞) is not a valid answer here because it is not physically meaningful in the context of this scenario. The charge at x = 0 generates an electric field that extends infinitely, but we are only considering the region of interest between the two charges. Hence, we conclude that there is no point where the potential is zero for x < 0 in this specific scenario.
It is worth noting that the point at x = -9.6 is an extraneous solution that arises when you try to solve the equation without considering the physical constraints of the scenario. So, it is important to take into account the physical context when interpreting the solutions of equations.
In summary, in the given scenario, there is no point where the potential is zero for x < 0. The point where the potential is zero is between the two charges, approximately x = 1.92 cm, as you correctly mentioned. Additionally, for x > 4.8 cm, there is no point where the potential is zero.