Two charges, both with charge +Q, have been placed on the x-axis at x=−a and x=a. Where will the electric field be zero?
At the origin (unstable equilibrium, the slightest y direction motion causes +E away) and at infinity in all directions
To find the point where the electric field is zero, we can consider the principle of superposition. According to this principle, the total electric field at any point in space is the vector sum of the electric fields due to individual charges.
In this case, we have two charges located at x = -a and x = a, both with charge +Q.
Let's first calculate the electric field at a general point P on the x-axis and denote it as E1 due to the charge at x = -a and E2 due to the charge at x = a.
The electric field due to a point charge can be calculated using Coulomb's law:
E1 = k * Q / r1²
E2 = k * Q / r2²
Where k is the electrostatic constant, Q is the magnitude of the charge, r1 is the distance between the charge at x = -a and point P, and r2 is the distance between the charge at x = a and point P.
Since we want to find the point where the electric field is zero, we need to set E1 + E2 = 0.
That can be written as:
k * Q / r1² + k * Q / r2² = 0
We can simplify this equation by multiplying both sides by r1² * r2²:
k * Q * r2² + k * Q * r1² = 0
Now, we can solve this equation to find the relationship between r1 and r2 when the electric field is zero. However, it's important to note that the solution will not result in a unique point, as there will be multiple points along the x-axis where the electric field is zero.
To obtain the specific points where the electric field is zero, we need additional information, such as the values of a, Q, and k.