A point charge of +1 µC is located at x =
−2 cm and a second point charge of −8 µC is located at x = 6 cm.
Where should a third charge of +1 µC be
placed so that the electric field at x = 0 cm is
zero?
Call the position of the third charge x.
Write the expression for the total E field.
E = k*10^-6*10^4[1/2^2 +8/6^2 -1/(x)^2]
Set it equal to zero.
Then solve for x.
1/x^2 = 1/4 + 8/36 = 17/36
x =sqrt(36/17)= 1.46 cm
To find the position where the electric field is zero at x = 0 cm, we need to calculate the net electric field at that point due to the two existing charges. The electric field due to a point charge is given by the formula:
Electric field (E) = k * q / r^2
Where:
- k is the electrostatic constant, approximately equal to 9 * 10^9 Nm^2/C^2
- q is the charge of the point charge
- r is the distance between the point charge and the point where the electric field is being calculated
Let's calculate the individual electric fields due to the +1 µC and -8 µC charges at x = 0 cm.
For the +1 µC charge at x = -2 cm:
r1 = 2 cm = 0.02 m
E1 = k * q / r1^2
For the -8 µC charge at x = 6 cm:
r2 = 6 cm = 0.06 m
E2 = k * q / r2^2
Now, since the electric field at x = 0 cm is the sum of the individual electric fields, we want the net electric field to be zero. Mathematically, this can be expressed as:
E_net = E1 + E2 = 0
Substituting the values of E1 and E2, we get:
k * q / 0.02^2 + k * q / 0.06^2 = 0
Now, we can solve this equation to find the value of q that satisfies this condition.
To find the position where a third charge of +1 µC should be placed so that the electric field at x = 0 cm is zero, we need to consider the net electric field at x = 0 cm due to the two existing charges.
Let's calculate the electric field at x = 0 cm due to the +1 µC charge and the -8 µC charge individually.
1. Electric field due to the +1 µC charge:
The electric field due to a point charge can be calculated using the formula:
E = k * q / r^2
Where:
- E is the electric field
- k is the electrostatic constant (9 × 10^9 Nm^2/C^2)
- q is the charge
- r is the distance from the point charge to the location where we want to find the electric field
Given that the +1 µC charge is at x = -2 cm (or -0.02 m), the distance r from this charge to x = 0 cm is:
r = x = -0.02 m
Substituting the values into the formula:
E1 = (9 × 10^9 Nm^2/C^2) * (1 × 10^-6 C) / (0.02 m)^2
2. Electric field due to the -8 µC charge:
Using the same formula and the distance from the -8 µC charge at x = 6 cm (or 0.06 m) to x = 0 cm:
r = x = 0.06 m
Substituting the values into the formula:
E2 = (9 × 10^9 Nm^2/C^2) * (-8 × 10^-6 C) / (0.06 m)^2
Now, to cancel out the electric field at x = 0 cm, the magnitudes of E1 and E2 should be equal.
Therefore, we can set:
|E1| = |E2|
(9 × 10^9 Nm^2/C^2) * (1 × 10^-6 C) / (0.02 m)^2 = (9 × 10^9 Nm^2/C^2) * (-8 × 10^-6 C) / (0.06 m)^2
Simplifying the equation further will give us the distance where the third charge should be placed:
(1 × 10^-6 C) / (0.02 m)^2 = (-8 × 10^-6 C) / (0.06 m)^2
(1 × 10^-6 C) * (0.06 m)^2 = (-8 × 10^-6 C) * (0.02 m)^2
0.00000036 C * 0.0036 m^2 = -0.00000016 C * 0.0004 m^2
1.296 × 10^-9 Cm^2 = -6.4 × 10^-11 Cm^2
As the left side of the equation is positive and the right side is negative, there is no real solution. Therefore, it is not possible to place a third charge of +1 µC at any position such that the electric field at x = 0 cm is zero when considering only the given charges.