Two point charges of equal magnitude Q are held a distance d apart. Consider only points on the line passing through both charges. (a) If the two charges have the same sign, find the location of all points (if there are any) at which (i) the potential (relative to infinity) is zero (is the electric field zero at these points?), and (ii) the electric field is zero (is the potential zero at these points?).

Verified step by step guidance

1. Understand the concept of electric potential and electric field. The electric potential at a point due to a point charge is given by \( V = \frac{kQ}{r} \), where \( k \) is Coulomb's constant, \( Q \) is the charge, and \( r \) is the distance from the charge. The electric field due to a point charge is given by \( E = \frac{kQ}{r^2} \).
2. For part (a)(i), consider the potential due to two charges of the same sign. The potential at a point on the line is the algebraic sum of the potentials due to each charge. Set up the equation \( V = \frac{kQ}{r_1} + \frac{kQ}{r_2} = 0 \), where \( r_1 \) and \( r_2 \) are the distances from the point to each charge. Solve for the location where this condition is satisfied.
3. Analyze whether the electric field is zero at the points where the potential is zero. The electric field is the vector sum of the fields due to each charge. Set up the equation \( E = \frac{kQ}{r_1^2} - \frac{kQ}{r_2^2} = 0 \) and solve for the location where this condition is satisfied. Compare this with the locations found in step 2.
4. For part (a)(ii), find the location where the electric field is zero. This occurs when the magnitudes of the electric fields due to each charge are equal and opposite. Set up the equation \( \frac{kQ}{r_1^2} = \frac{kQ}{r_2^2} \) and solve for the location where this condition is satisfied.
5. Determine whether the potential is zero at the points where the electric field is zero. Use the equation from step 2 to check if the potential is zero at the locations found in step 4. Discuss the relationship between the electric field and potential at these points.

Key Concepts

Electric Potential

Electric potential at a point in space is the work done per unit charge in bringing a positive test charge from infinity to that point. For two like charges, the potential at a point is the algebraic sum of potentials due to each charge. Points where the potential is zero are equidistant from both charges, but the electric field may not be zero at these points.

Electric Field

The electric field is a vector field representing the force per unit charge experienced by a test charge at a point in space. For two like charges, the field is zero at a point where the forces due to each charge cancel each other out. This occurs at the midpoint between the charges, but the potential is not necessarily zero at these points.

Superposition Principle

The superposition principle states that the net electric field or potential at a point is the vector sum of fields or potentials due to individual charges. This principle is crucial for calculating the potential and field at any point on the line between two charges, allowing us to determine where these quantities are zero.