Question

Which of the following situations produces the largest net force on the charge \(Q\) ? a) Charge \(Q=1 \mathrm{C}\) is \(1 \mathrm{~m}\) from a charge of \(-2 \mathrm{C}\). b) Charge \(Q=1 \mathrm{C}\) is \(0.5 \mathrm{~m}\) from a charge of \(-1 \mathrm{C}\) c) Charge \(Q=1 \mathrm{C}\) is halfway between a charge of \(-1 \mathrm{C}\) and a charge of \(1 \mathrm{C}\) that are \(2 \mathrm{~m}\) apart. d) Charge \(Q=1 \mathrm{C}\) is halfway between two charges of \(-2 \mathrm{C}\) that are \(2 \mathrm{~m}\) apart. e) Charge \(Q=1 \mathrm{C}\) is a distance of \(2 \mathrm{~m}\) from a charge of \(-4 \mathrm{C}\).

Short answer

Answer: The largest net force is in situation b) with a magnitude of 35.96 x 10^9 N.

Step-by-step solution

Compute force for situation a)

For the situation a), Q = 1C, 1m from a charge of -2C. We have: q1 = 1 C q2 = -2 C r = 1 m Using the Coulomb's Law formula, we get the force Fa: \(Fa = k \frac{q1 \cdot q2}{r^2} = 8.99 \times 10^9\frac{1 \times (-2)}{1^2} = -17.98 \times 10^9 N\)

Compute force for situation b)

For the situation b), Q = 1C, 0.5m from a charge of -1C. We have: q1 = 1 C q2 = -1 C r = 0.5 m Using the Coulomb's Law formula, we get the force Fb: \(Fb = k \frac{q1 \cdot q2}{r^2} = 8.99 \times 10^9\frac{1 \times (-1)}{0.5^2} = -35.96 \times 10^9 N\)

Compute vector forces for situation c)

For the situation c), Q = 1C is halfway between a charge of -1C and a charge of 1C that are 2 m apart. First, find the distance to each charge: 1 m. Next, find the force caused by the -1C charge and the force caused by the +1C charge. Due to symmetry, these forces will have opposite directions and will cancel each other out. Thus, the net force is 0 N for this situation.

Compute vector forces for situation d)

For the situation d), Q = 1C is halfway between two charges of -2C that are 2 m apart. First, find the distance to each charge: 1 m. Calculate the force caused by each -2C charge: \(Fd1 = k \frac{1 \cdot (-2)}{1^2} = -17.98 \times 10^9 N\) Because of symmetry, the two forces have the same magnitude but act in opposite directions. The net force Fd: \(Fd = 2 \times |-17.98 \times 10^9 N| = 35.96 \times 10^9 N\)

Compute force for situation e)

For the situation e), Q = 1C, 2m from a charge of -4C. We have: q1 = 1 C q2 = -4 C r = 2 m Using the Coulomb's Law formula, the force Fe: \(Fe = k \frac{q1 \cdot q2}{r^2} = 8.99 \times 10^9\frac{1 \times (-4)}{2^2} = -17.98 \times 10^9 N\) Now we have all net forces for each situation: Fa = -17.98 x 10^9 N, Fb = -35.96 x 10^9 N, Fc = 0 N, Fd = 35.96 x 10^9 N, Fe = -17.98 x 10^9 N. The largest net force will have the greatest magnitude, which is Fb in situation b) with a magnitude of 35.96 x 10^9 N.

Key concepts

Electric Force

Electric force is a fundamental force that arises from the interaction between charged particles. This force is described by Coulomb's Law, a principle foundational in electrostatics. It states that the magnitude of the electric force between two point charges is directly proportional to the product of the amounts of the charges and inversely proportional to the square of the distance between them. This can be mathematically expressed as:\[ F = k \frac{|q_1 \cdot q_2|}{r^2} \]where:

• \( F \) is the electric force,
• \( q_1 \) and \( q_2 \) are the magnitudes of the charges,
• \( r \) is the distance between the charges,
• \( k \) is the Coulomb's constant (approximately \( 8.99 \times 10^9 \mathrm{N \cdot m^2/C^2} \)).

The electric force can be attractive or repulsive. An attractive force occurs when the charges have opposite signs, while a repulsive force happens when they have like signs. This concept is pivotal for understanding electric interactions in various situations.

Charge Interaction

Charge interaction refers to the way charged objects interact with each other through forces that can be either attractive or repulsive. This is rooted in the fundamental property of electric charges, where:

• Like charges repel each other.
• Opposite charges attract each other.

This interaction is quantified using Coulomb's Law, as previously mentioned, which allows us to calculate the force based on the amount of charge and the distance between them. The direction of the force is crucial when considering resulting movements or balance of forces.
For example, in scenario c) of the original exercise, charge interaction proved a net force of zero because the forces from two opposite charges canceled each other out. Understanding these interactions is critical in predicting the behavior of charge systems and determining net forces as in various configurations.

Distance Dependence in Forces

The relationship between distance and the magnitude of electric force is another critical aspect to understand. According to Coulomb's Law, the force between two charges diminishes as the distance between them increases.
The inverse square law nature of this relationship means doubling the distance between charges will decrease the force by a factor of four (\( r^2 \)). This has significant implications:

• Closer proximity results in stronger interactive forces.
• Moving charges further apart significantly weakens the force between them.

For instance, in situation b) of the exercise, reducing the distance between charges resulted in the greatest force magnitude, demonstrating how influential the distance parameter is in determining force strength. Recognizing the impact of changing distances helps one to predict how altering physical arrangements in charge systems affects the net forces observed.