Two 1.0 g spheres are charged equally and placed 2.0 cm apart. When released, they begin to accelerate at 150 m/s². What is the magnitude of the charge on each sphere?

Verified step by step guidance

1. Convert the given masses and distance into SI units. The mass of each sphere is 1.0 g, which is equivalent to 0.001 kg. The distance between the spheres is 2.0 cm, which is equivalent to 0.02 m.
2. Use Newton's second law of motion, \( F = ma \), to calculate the force acting on each sphere. Here, \( m = 0.001 \; \text{kg} \) and \( a = 150 \; \text{m/s}^2 \). Substitute these values into the equation to find the force \( F \).
3. Recognize that the force between the two charged spheres is due to Coulomb's law, which is given by \( F = \frac{k q^2}{r^2} \), where \( k = 8.99 \times 10^9 \; \text{N·m}^2/\text{C}^2 \) is Coulomb's constant, \( q \) is the charge on each sphere, and \( r = 0.02 \; \text{m} \) is the distance between the spheres.
4. Rearrange Coulomb's law to solve for \( q \): \( q = \sqrt{\frac{F r^2}{k}} \). Substitute the values of \( F \), \( r \), and \( k \) into this equation.
5. Simplify the expression to calculate the magnitude of the charge \( q \) on each sphere. Ensure that the units are consistent throughout the calculation.

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 relationship is crucial for calculating the force acting on the spheres due to their charges.

Newton's Second Law of Motion

Newton's Second Law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. This principle allows us to relate the force experienced by the charged spheres to their acceleration, enabling the calculation of the force due to the electrostatic interaction.

Acceleration

Acceleration is the rate of change of velocity of an object. In this context, it indicates how quickly the spheres are speeding up due to the electrostatic force. Understanding acceleration is essential for applying Newton's Second Law to find the net force acting on the spheres, which can then be used to determine the charge on each sphere.