Tutorial Questions on Electrostatics, Charge Transfer, and Particle Dynamics

Calculation of Distance for Specified Electrostatic Force

The fundamental problem in electrostatics as presented in question +3 SSM requires determining the spatial separation between two known point charges based on a given force magnitude. Specifically, the scenario involves a point charge $q_1 = 26.0\,\text{C}$ and a point charge $q_2 = -47.0\,\text{C}$. The objective is to calculate the distance $r$ required for the electrostatic force between these two charges to exhibit a magnitude of exactly $5.70\,\text{N}$. This application utilizes Coulomb's Law, which states that the force $F$ is equal to the product of the electrostatic constant $k$ (approximately $8.99 \times 10^9\,\text{N}\cdot\text{m}^2\cdot\text{C}^{-2}$) and the magnitudes of the two charges, divided by the square of the distance between them: $F = k \cdot \frac{|q_1 \cdot q_2|}{r^2}$. To isolate the distance, the formula is restructured as $r = \sqrt{\frac{k \cdot |q_1 \cdot q_2|}{F}}$.

Analysis of Charge Transfer in Typical Lightning Bolt Return Strokes

Question 4 25 focuses on the quantitative measurement of electrical charge transferred during atmospheric electrical discharges. In the context of a typical lightning bolt, the return stroke is a high-current event. The transcript specifies a current magnitude of $I = 2.5 \times 10^4\,\text{A}$ (where the notation ' represents the power of four) that exists for a temporal duration of $t = 20\,\text{us}$. To find the total charge transferred during this event, one must use the relationship between current and charge, where current is defined as the rate of flow of charge over time: $Q = I \times t$. When performing this calculation, the time must be appropriately treated as $20 \times 10^{-6}\,\text{s}$ to ensure that the resulting charge is measured in Coulombs.

Magnitude of Electrostatic Force Between Two Static Point Charges

Problem -5 details a scenario involving two charged particles to determine the resulting magnitude of the electrostatic force acting between them. The first particle carries a charge defined as $q_1 = +3.00 \times 10^{\circ}\,\text{C}$, and it is situated at a distance of $12.0\,\text{cm}$ (equivalent to $0.120\,\text{m}$) from a second particle with a charge of $q_2 = -1.50 \times 10\,\text{C}$. The solution requires the direct application of Coulomb's Law: $F = k \cdot \frac{|q_1 \cdot q_2|}{r^2}$. Due to the opposing signs of the charges—one positive and one negative—the resulting force is attractive in nature, though the calculation specifically seeks the magnitude of this force. The distance must be squared in the denominator to reflect the inverse-square law governing electrostatic interactions.

Determination of Mass and Charge via Particle Acceleration and Kinematics

Question +6 ILW explores the dynamic interactions of two equally charged particles when released from rest. The particles are initially held at a separation distance of $d = 3.2 \times 10^{-3}\,\text{m}$. Upon release, the particles experience mutual electrostatic repulsion (given they are equally charged), resulting in an observed initial acceleration for the first particle of $a_1 = 7.0\,\text{m/s}^2$ and for the second particle of $a_2 = 9.0\,\text{m/s}^2$. The mass of the first particle is given as $m_1 = 6.3 \times 10^7\,\text{kg}$.

This problem is solved in two distinct parts. In part (a), the mass of the second particle ($m_2$) is found by applying Newton's Third Law and Second Law. Since the electrostatic force exerted by particle 1 on particle 2 is equal in magnitude to the force exerted by particle 2 on particle 1 ($F_{12} = F_{21}$), it follows that $m_1 \cdot a_1 = m_2 \cdot a_2$. Solving for the unknown mass yields $m_2 = \frac{m_1 \cdot a_1}{a_2}$. In part (b), the magnitude of the charge on each particle ($q$) is determined. Because the particles are equally charged, the force is $F = k \cdot \frac{q^2}{d^2}$. By setting this equal to the product of mass and acceleration ($m_1 \cdot a_1$), the charge can be calculated using the expression $q = \sqrt{\frac{m_1 \cdot a_1 \cdot d^2}{k}}$.

Component Analysis of Net Electrostatic Force in Square Particle Geometries

Problem 5, referencing Figure 21-24, involves a multi-particle system where four particles are arranged at the corners of a square. The geometry is defined by a side length $a = 5.0\,\text{cm}$. The charges of the particles are specified as $q_1 = -q_2 = 100\,\text{nC}$ and $q_3 = q_4 = 200\,\text{nC}$. The primary goal is to find the vector components of the net electrostatic force acting on particle 3.

To determine the (a) x and (b) y components of the net force on particle 3, the Principle of Superposition must be applied, meaning the net force is the vector sum of the individual forces exerted by particles 1, 2, and 4. The distances must be calculated based on the square geometry: particles adjacent to particle 3 are at distance $a$, while the particle at the opposite corner (particle 1) is at the diagonal distance of $a\sqrt{2}$. The calculation involves finding the magnitude of each individual force—$F_{31}$, $F_{32}$, and $F_{34}$—using Coulomb's Law and then decomposing these into their respective horizontal and vertical components based on the coordinate system of the square. Any charges provided in nanocoulombs (nC) must be converted to Coulombs by multiplying by $10^{-9}$.