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give the calculation step by step as like as full answer in 2 steps and explanation block in every steps. And final answer summary Definition of Parameters Solutions are reviewed by our quality team mainly based on 6 parameters: Relevancy, Completeness, Accuracy, Clarity, Structure, and Voice. Relevancy: This indicates how closely a solution aligns with the question asked. It evaluates whether the solution addresses the core aspects of the question and provides appropriate information. Completeness: This indicates how thoroughly the solution addresses various asks in the question. It evaluates whether the solution has all essential information like calculations, diagrams, procedural steps, & explanatory statements in the solution. Accuracy: It refers to how correct the provided solution is for the given question/s. This evaluates the correctness of the information/concept/method followed in the solution. Clarity: Clarity refers to how easily the solution can be understood by the student. This evaluates the language complexity, readability, and redundancy of a solution. Structure: Structure refers to how the solution is organized and presented in relation to the requirements of the question. This evaluates the necessity of steps and their effectiveness in helping the student grasp how to arrive at the solution. Voice: This refers to the language and tone of the solution. This primarily evaluates whether a solution sounds human or overly non-human (bot-like). don't use we, then,:.A positive charge q is fixed at point (3,4) and a negative charge —q is fixed at point (3,0). Determine the net electric force Frat acting on a negative test charge —Q at the origin (0,0) in terms of the given quantities and the permittivity of free space €g. Enter your expression using ij unit vector notation and rational coefficients.

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give the calculation step by step as like as full answer in 2 steps and explanation block in every steps. And final answer summary Definition of Parameters Solutions are reviewed by our quality team mainly based on 6 parameters: Relevancy, Completeness, Accuracy, Clarity, Structure, and Voice. Relevancy: This indicates how closely a solution aligns with the question asked. It evaluates whether the solution addresses the core aspects of the question and provides appropriate information. Completeness: This indicates how thoroughly the solution addresses various asks in the question. It evaluates whether the solution has all essential information like calculations, diagrams, procedural steps, & explanatory statements in the solution. Accuracy: It refers to how correct the provided solution is for the given question/s. This evaluates the correctness of the information/concept/method followed in the solution. Clarity: Clarity refers to how easily the solution can be understood by the student. This evaluates the language complexity, readability, and redundancy of a solution. Structure: Structure refers to how the solution is organized and presented in relation to the requirements of the question. This evaluates the necessity of steps and their effectiveness in helping the student grasp how to arrive at the solution. Voice: This refers to the language and tone of the solution. This primarily evaluates whether a solution sounds human or overly non-human (bot-like). don't use we, then,:.Uploaded ImageA positive charge q is fixed at point (3,4) and a negative charge —q is fixed at point (3,0). Determine the net electric force Frat acting on a negative test charge —Q at the origin (0,0) in terms of the given quantities and the permittivity of free space €g. Enter your expression using ij unit vector notation and rational coefficients.

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Asked by: sai
Created at: 2025-06-09 00:43:07

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**Step 1: Calculate the Force on -Q Due to Each Source Charge** The electric force between two point charges is given by Coulomb’s Law: \[ \vec{F} = \frac{1}{4\pi \epsilon_0} \frac{q_1 q_2}{r^2} \hat{r} \] - Point A: Positive charge \(q\) at \((3,4)\) - Point B: Negative charge \(-q\) at \((3,0)\) - Test charge: \(-Q\) at \((0,0)\) **Force due to charge at (3,4):** - Displacement vector from (3,4) to (0,0): \(\vec{r}_1 = (0-3)\hat{i} + (0-4)\hat{j} = -3\hat{i} - 4\hat{j}\) - Distance: \(r_1 = \sqrt{(-3)^2 + (-4)^2} = 5\) - Unit vector: \(\hat{r}_1 = \frac{-3\hat{i} - 4\hat{j}}{5}\) - The force magnitude: \(F_1 = \frac{1}{4\pi\epsilon_0} \frac{q \cdot (-Q)}{(5)^2}\) - Direction: along \(\hat{r}_1\) So, \[ \vec{F}_1 = \frac{1}{4\pi\epsilon_0} \frac{-qQ}{25} \left(\frac{-3\hat{i} - 4\hat{j}}{5}\right) = \frac{1}{4\pi\epsilon_0} \frac{-qQ}{125} (-3\hat{i} - 4\hat{j}) \] **Explanation:** The force is attractive because the test charge and the source charge have opposite signs. The direction is from the test charge to the source charge. --- **Force due to charge at (3,0):** - Displacement vector: \(\vec{r}_2 = (0-3)\hat{i} + (0-0)\hat{j} = -3\hat{i}\) - Distance: \(r_2 = 3\) - Unit vector: \(\hat{r}_2 = -\hat{i}\) - The force magnitude: \(F_2 = \frac{1}{4\pi\epsilon_0} \frac{(-q) \cdot (-Q)}{3^2} = \frac{1}{4\pi\epsilon_0} \frac{qQ}{9}\) - Direction: along \(\hat{r}_2\) So, \[ \vec{F}_2 = \frac{1}{4\pi\epsilon_0} \frac{qQ}{9} (-\hat{i}) \] **Explanation:** Both charges are negative, so the force is repulsive. The force on \(-Q\) at the origin is directed away from \(-q\) at (3,0), i.e., in the negative \(x\)-direction. --- **Step 2: Add the Vector Forces** Sum the \( \hat{i} \) and \( \hat{j} \) components from both forces: \[ \vec{F}_{\text{net}} = \vec{F}_1 + \vec{F}_2 \] Expanding the \( \hat{i} \) component: - \( \vec{F}_1 \) gives \( \frac{1}{4\pi\epsilon_0} \frac{-qQ}{125} (-3) = \frac{3qQ}{500\pi\epsilon_0} \) in \( \hat{i} \) - \( \vec{F}_2 \) gives \( \frac{1}{4\pi\epsilon_0} \frac{qQ}{9}(-1) = -\frac{qQ}{36\pi\epsilon_0} \) in \( \hat{i} \) \( \hat{j} \) component: - Only \( \vec{F}_1 \) contributes: \( \frac{1}{4\pi\epsilon_0} \frac{-qQ}{125} (-4) = \frac{4qQ}{500\pi\epsilon_0} \) So, \[ \vec{F}_{\text{net}} = \left(\frac{3qQ}{500\pi\epsilon_0} - \frac{qQ}{36\pi\epsilon_0}\right)\hat{i} + \frac{4qQ}{500\pi\epsilon_0}\hat{j} \] Write with rational coefficients: \[ \frac{3}{500} - \frac{1}{36} = \frac{3 \times 36 - 1 \times 500}{500 \times 36} = \frac{108 - 500}{18000} = -\frac{392}{18000} = -\frac{49}{2250} \] So, \[ \vec{F}_{\text{net}} = -\frac{49qQ}{2250\pi\epsilon_0}\hat{i} + \frac{4qQ}{500\pi\epsilon_0}\hat{j} \] **Explanation:** The net force is found by adding the contributions from each source charge, using unit vector notation and simplifying the coefficients to rational form as requested. --- **Final Answer Summary** **Net electric force on the test charge is:** \[ \boxed{ \vec{F}_{\text{net}} = -\frac{49qQ}{2250\pi\epsilon_0}\,\hat{i} + \frac{4qQ}{500\pi\epsilon_0}\,\hat{j} } \] This vector is expressed in \( \hat{i}, \hat{j} \) unit notation and uses rational coefficients, in terms of \( q, Q, \epsilon_0 \).

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