Four small spheres, each charged to +15 nC, form a square 2.0 cm on each side. From far away, a proton is shot toward the square along a line perpendicular to the square and passing through its center. What minimum initial speed does the proton need to pass through the square of charges?

Verified step by step guidance

1. Identify the forces acting on the proton: The proton is positively charged, so it will experience a repulsive electric force from each of the four positively charged spheres. The net force will act along the line of motion due to symmetry.
2. Calculate the electric field at the center of the square: Use the formula for the electric field due to a point charge, \( E = \frac{kq}{r^2} \), where \( k \) is Coulomb's constant, \( q \) is the charge of each sphere, and \( r \) is the distance from the charge to the center of the square. For each sphere, \( r = \sqrt{(\frac{a}{2})^2 + (\frac{a}{2})^2} \), where \( a \) is the side length of the square.
3. Determine the total electric field at the center: Since the electric field vectors from all four charges point radially outward and have the same magnitude, their components along the line perpendicular to the square will add up. Use vector addition to find the net electric field along the proton's path.
4. Relate the electric field to the force on the proton: The force on the proton is given by \( F = q_p E \), where \( q_p \) is the charge of the proton. This force will act to decelerate the proton as it approaches the square.
5. Apply energy conservation to find the minimum initial speed: The proton's initial kinetic energy, \( \frac{1}{2}mv^2 \), must be sufficient to overcome the work done by the electric force as it moves through the electric field. Set the initial kinetic energy equal to the electric potential energy at the center of the square, \( U = q_p V \), where \( V \) is the electric potential at the center due to the four charges. Solve for the initial speed \( v \).

Key Concepts

Coulomb's Law

Coulomb's Law describes the electrostatic force between two charged objects. It states that the force is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. This principle is crucial for calculating the forces acting on the proton as it approaches the charged spheres.

Electric Field

The electric field is a vector field surrounding charged particles that exerts a force on other charges within the field. The strength and direction of the electric field depend on the charge distribution. Understanding the electric field created by the four charged spheres is essential to determine the force acting on the proton as it moves toward the square.

Kinetic and Potential Energy

Kinetic energy is the energy of an object due to its motion, while potential energy is the stored energy based on its position in a field, such as an electric field. As the proton approaches the charged spheres, it converts kinetic energy into electric potential energy. Analyzing the balance between these energies helps determine the minimum speed required for the proton to pass through the square without being repelled.