two point charges lie along the x "-axis." a charge of q1 = −9.0 μc is at x = 6.0 m, and a charge of q2 = −8.0 μc is at x = −4.0 m. locate the point (other than infinity) at which the total electric field is zero.
To find the point (other than infinity) where the total electric field is zero, we need to calculate the electric fields created by the two point charges separately and then determine the position at which they cancel each other out.
Given:
Charge q1 = -9.0 μC (microcoulombs) at x = 6.0 m
Charge q2 = -8.0 μC (microcoulombs) at x = -4.0 m
Let's calculate the electric fields created by each charge:
1. Electric field created by q1:
Electric field due to a point charge can be calculated using the formula:
E1 = k * (q1 / r1^2)
Where k is the electrostatic constant (k = 8.99 x 10^9 Nm^2/C^2),
q1 is the magnitude of charge q1,
and r1 is the distance from q1 to the point where we want to find the electric field.
The distance between q1 and the point we want to find the electric field is x1 = x - 6.0 m.
So, r1 = |x1|.
2. Electric field created by q2:
Similar to q1, the electric field due to q2 can be calculated as:
E2 = k * (q2 / r2^2)
Where q2 is the magnitude of charge q2,
and r2 is the distance from q2 to the point where we want to find the electric field.
The distance between q2 and the point we want to find the electric field is x2 = x + 4.0 m.
So, r2 = |x2|.
Now, let's set up an equation to find the position x at which the total electric field is zero:
E1 + E2 = 0
k * (q1 / r1^2) + k * (q2 / r2^2) = 0
Substituting the values:
8.99 x 10^9 * (-9.0 x 10^-6) / |x1|^2 + 8.99 x 10^9 * (-8.0 x 10^-6) / |x2|^2 = 0
Simplifying the equation, we have:
-9.0 / |x1|^2 - 8.0 / |x2|^2 = 0
To locate the point at which the total electric field is zero, we need to solve this equation for x.
To find the point at which the total electric field is zero, we need to consider the electric fields created by both charges and find the point where their vector sum cancels out.
The electric field created by a point charge is given by Coulomb's law:
E = k * q / r^2
Where E is the electric field, k is the Coulomb's constant (9 x 10^9 Nm^2/C^2), q is the charge, and r is the distance from the charge to the point of interest.
Let's first find the electric field created by q1 at any point along the x-axis. Since q1 is negative, its electric field points towards the charge. Thus, at any point x on the x-axis, the electric field created by q1 is:
E1 = -k * q1 / (x - 6)^2
Similarly, let's find the electric field created by q2 at any point along the x-axis. Since q2 is negative, its electric field also points towards it. Thus, at any point x on the x-axis, the electric field created by q2 is:
E2 = -k * q2 / (x + 4)^2
To find the point where the total electric field is zero, we need to add the electric fields created by q1 and q2 and set the sum to zero:
E1 + E2 = 0
Substituting the expressions for E1 and E2:
-k * q1 / (x - 6)^2 - k * q2 / (x + 4)^2 = 0
Now, we can solve this equation to find the value of x where the total electric field is zero.