A charge Q has to be divided between two solid spheres of radius 'R' which are at distance d from each other (d >> R). What should be the value of charge, which we should place on spheres, so that the force of attraction between them is maximum?

Explanation

To find the value of charge that should be placed on each sphere so that the force of attraction between them is maximum, we can use the principle of superposition. Let's denote the charge on each sphere as q. The force of attraction between two point charges q1 and q2 separated by a distance r is given by Coulomb's Law: F = k * |q1 * q2| / r^2, where F is the force of attraction, k is Coulomb's constant (approximately 8.99 x 10^9 N m^2/C^2), and r is the distance between the charges. In this scenario, each sphere can be considered as a point charge located at its center. The distance between the centers of the two spheres is d. Given that the distance d is much greater than the radius R (d >> R), we can assume that the two spheres are effectively point charges for the purpose of this calculation. Let's consider the force of attraction between the two spheres: Force = k * |q * q| / d^2. Since we want to find the value of charge q that maximizes the force of attraction, we need to take the derivative of the force with respect to q and set it equal to zero to find the critical points. Then, we can determine whether the critical point corresponds to a maximum or a minimum value. Let's calculate it step by step: (since each sphere has a charge q). Take the derivative with respect to q: d(Force)/dq = 4 * k * q / d^2. Set the derivative equal to zero to find the critical point: 4 * k * q / d^2 = 0. The above equation simplifies to q = 0. This means that the derivative is zero when the charge on each sphere is zero. However, this is not the point of maximum force since it corresponds to no charge on the spheres. To find the maximum force, we need to look at the endpoints of the possible charge values. Since charge cannot be negative, the endpoints are q = 0 and q = Q (the total charge). So, we need to calculate the force at these endpoints. For q = 0, Force = 2 * k * 0^2 / d^2 = 0. For q = Q, Force = 2 * k * Q^2 / d^2. Since Force is directly proportional to Q^2, to maximize the force, we need to maximize Q. Therefore, all of the charge Q should be placed on each of the spheres to maximize the force of attraction between them.