Question

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?

Accepted Answer

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. 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. 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. 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. 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 $$ v .$$

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?

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Concetti chiave

Coulomb's Law

Electric Field

Kinetic and Potential Energy