Math Problem Statement
Gegeben sei ein Kraftfeld F = (-ky, -kx) a) Ist dieses Kraftfeld konservativ? b) Skizzieren Sie das Feld durch kleine Pfeile, deren Länge proportional zu |F| ist, in den Gitterpunkten {(x, y) | x = 0, ±1, ±2, ±3; y = 0, ±1, ±2 ±3}. c) Berechnen Sie durch Integration des Kraftfeldes die potentielle Energie E_pot(x, y), und machen Sie die Probe durch Differenzieren. Wie groß ist E_pot(P1) - E_pot(P2) für P1 = (-3, -3) und P2 = (3, 1)? d) Skizzieren Sie die Äquipotentiallinien für E_pot(x, y) = 0, ±2k, ±4k, ±6k, ±8k für |x| ≤ 3, |y| ≤ 3.
Solution
Let's address the problem step by step:
Part (a): Is the field conservative?
The vector field is given as .
To check if it is conservative, compute the curl of the field:
\begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ -k y & -k x & 0 \end{vmatrix}$$ Expanding the determinant: $$\text{curl}(\vec{F}) = \hat{i} \left(\frac{\partial}{\partial y}(0) - \frac{\partial}{\partial z}(-kx)\right) - \hat{j} \left(\frac{\partial}{\partial x}(0) - \frac{\partial}{\partial z}(-ky)\right) + \hat{k} \left(\frac{\partial}{\partial x}(-kx) - \frac{\partial}{\partial y}(-ky)\right)$$ Simplify each term: - $$\frac{\partial}{\partial x}(-kx) = -k$$, - $$\frac{\partial}{\partial y}(-ky) = -k$$. Thus: $$\text{curl}(\vec{F}) = \hat{k}(-k - (-k)) = \hat{k}(0) = 0.$$ Since the curl is zero, the field is **conservative**. --- ### **Part (b): Sketch the vector field** The length of each vector is proportional to the magnitude of $$\vec{F}$$, which is: $$|\vec{F}| = \sqrt{(-ky)^2 + (-kx)^2} = k\sqrt{x^2 + y^2}.$$ For $$(x, y)$$ on the grid points $$\{(x, y) \mid x = 0, \pm1, \pm2, \pm3; y = 0, \pm1, \pm2, \pm3\}$$, sketch small arrows pointing in the direction of the field $$(-ky, -kx)$$, whose length is proportional to $$|\vec{F}|$$. --- ### **Part (c): Compute $$E_{\text{pot}}(x, y)$$ and check using differentiation** Since the field is conservative, the potential energy $$E_{\text{pot}}(x, y)$$ can be computed as: $$E_{\text{pot}}(x, y) = -\int \vec{F} \cdot d\vec{r},$$ where $$d\vec{r} = (dx, dy)$$. Compute $$E_{\text{pot}}$$: $$E_{\text{pot}}(x, y) = -\int (-ky) dx - \int (-kx) dy.$$ Solve each integral: - $$\int (-ky) dx = -kyx + C_1(y)$$, - $$\int (-kx) dy = -kxy + C_2(x)$$. Combine terms: $$E_{\text{pot}}(x, y) = -kxy + C,$$ where $$C$$ is an arbitrary constant. **Verification:** Differentiate $$E_{\text{pot}}(x, y) = -kxy$$ to recover $$\vec{F}$$: - $$\frac{\partial E_{\text{pot}}}{\partial x} = -ky$$ matches $$F_x = -ky$$, - $$\frac{\partial E_{\text{pot}}}{\partial y} = -kx$$ matches $$F_y = -kx$$. Thus, the computation is verified. **Compute $$E_{\text{pot}}(P_1) - E_{\text{pot}}(P_2)$$:** For $$P_1 = (-3, -3)$$ and $$P_2 = (3, 1)$$: $$E_{\text{pot}}(P_1) = -k(-3)(-3) = -9k,$$ $$E_{\text{pot}}(P_2) = -k(3)(1) = -3k.$$ Difference: $$E_{\text{pot}}(P_1) - E_{\text{pot}}(P_2) = -9k - (-3k) = -6k.$$ --- ### **Part (d): Sketch equipotential lines** Equipotential lines for $$E_{\text{pot}}(x, y) = \text{const}$$ are given by: $$-kxy = \text{const}.$$ For $$E_{\text{pot}} = 0, \pm2k, \pm4k, \pm6k, \pm8k$$, plot hyperbolas within $$|x| \leq 3, |y| \leq 3$$. --- Do you want me to sketch the plots or provide further detailed calculations? ### **5 Follow-Up Questions:** 1. Can you explain why a field being conservative relates to the potential function? 2. How do you determine the direction of arrows in a vector field plot? 3. Why does the curl being zero imply conservativeness? 4. What are practical applications of equipotential lines in physics? 5. How does the magnitude of $$\vec{F}$$ affect the arrow lengths in part (b)? **Tip:** Always verify the conservative nature of a field before computing a potential function.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Vector fields
Conservative fields
Potential energy
Differentiation
Integration
Formulas
F = (-ky, -kx)
|F| = sqrt((-ky)^2 + (-kx)^2)
E_pot(x, y) = -∫ F · dr
Theorems
A vector field is conservative if its curl is zero.
Potential energy is related to the negative of the work done by a force.
Suitable Grade Level
Undergraduate physics or engineering courses
Related Recommendation
Vector Field Conservativeness and Line Integral Calculation
Determining if a Vector Field is Conservative and Finding a Potential Function
Analyzing Vector Field for Conservative Properties and Finding Potential Function
Determine if a Vector Field is Conservative and Find Its Potential Function
Find Potential Function of a Conservative Vector Field