College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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Question
![**Problem Statement:**
Given the potential function \( V = 5.6x^3yz + 7.5xy^4 \), calculate the x-component of the electric field at the point (6.9, 6.9, 13.36).
**Instructions:**
Round your answer to 2 decimal places.
---
**Explanation:**
1. **Electric Field and Potential Relationship:** The electric field \(\mathbf{E}\) is related to the electric potential \(V\) by the negative gradient:
\[ \mathbf{E} = -\nabla V \]
2. **X-Component of the Electric Field:**
- To find the x-component of the electric field \(\mathbf{E_x}\), we need to take the partial derivative of \(V\) with respect to \(x\):
\[ E_x = -\frac{\partial V}{\partial x} \]
3. **Partial Derivative Calculation:**
- Start with \( V = 5.6x^3yz + 7.5xy^4 \).
- Differentiate with respect to \(x\) while treating \(y\) and \(z\) as constants:
\[ \frac{\partial V}{\partial x} = \frac{\partial}{\partial x} (5.6x^3yz) + \frac{\partial}{\partial x} (7.5xy^4) \]
\[ = 3 \cdot 5.6x^2yz + 7.5y^4 \]
\[ = 16.8x^2yz + 7.5y^4 \]
4. **Evaluate at the Point (6.9, 6.9, 13.36):**
\[ \frac{\partial V}{\partial x} \bigg|_{(6.9, 6.9, 13.36)} = 16.8 \cdot (6.9)^2 \cdot 6.9 \cdot 13.36 + 7.5 \cdot (6.9)^4 \]
5. **Compute the Value:**
\[ \frac{\partial V}{\partial x} \bigg|_{(6.9, 6.9, 13.36)} \approx \text{(](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa023f3c3-3186-4c61-aa74-30f5825e9d0f%2Fe8020109-99c6-4fb2-8e45-0528081037d7%2F2a2wsr.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Given the potential function \( V = 5.6x^3yz + 7.5xy^4 \), calculate the x-component of the electric field at the point (6.9, 6.9, 13.36).
**Instructions:**
Round your answer to 2 decimal places.
---
**Explanation:**
1. **Electric Field and Potential Relationship:** The electric field \(\mathbf{E}\) is related to the electric potential \(V\) by the negative gradient:
\[ \mathbf{E} = -\nabla V \]
2. **X-Component of the Electric Field:**
- To find the x-component of the electric field \(\mathbf{E_x}\), we need to take the partial derivative of \(V\) with respect to \(x\):
\[ E_x = -\frac{\partial V}{\partial x} \]
3. **Partial Derivative Calculation:**
- Start with \( V = 5.6x^3yz + 7.5xy^4 \).
- Differentiate with respect to \(x\) while treating \(y\) and \(z\) as constants:
\[ \frac{\partial V}{\partial x} = \frac{\partial}{\partial x} (5.6x^3yz) + \frac{\partial}{\partial x} (7.5xy^4) \]
\[ = 3 \cdot 5.6x^2yz + 7.5y^4 \]
\[ = 16.8x^2yz + 7.5y^4 \]
4. **Evaluate at the Point (6.9, 6.9, 13.36):**
\[ \frac{\partial V}{\partial x} \bigg|_{(6.9, 6.9, 13.36)} = 16.8 \cdot (6.9)^2 \cdot 6.9 \cdot 13.36 + 7.5 \cdot (6.9)^4 \]
5. **Compute the Value:**
\[ \frac{\partial V}{\partial x} \bigg|_{(6.9, 6.9, 13.36)} \approx \text{(
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