Question

Consider a two-dimensional force field
\(\overrightarrow{F}(x,y) = (5x^2 + ay^2 + bxy)\hat{x} + (4x^2 + 4xy + y^2) \hat{y}\).
If the force field is conservative, then the values of a and b are

• A. a = 2 and b = 4
• B. a = 2 and b = 8
• C. a = 4 and b = 2
• D. a = 8 and b = 2

Answer 1

The Correct Option is B

To determine the values of \( a \) and \( b \) for which the given force field is conservative, we start by recalling that for a two-dimensional vector field \(\overrightarrow{F}(x, y) = P(x, y) \hat{x} + Q(x, y) \hat{y}\), to be conservative, it must satisfy:

\(\frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x}\)

Given the force field:

\(\overrightarrow{F}(x,y) = (5x^2 + ay^2 + bxy)\hat{x} + (4x^2 + 4xy + y^2) \hat{y}\)

Here, \( P(x, y) = 5x^2 + ay^2 + bxy \) and \( Q(x, y) = 4x^2 + 4xy + y^2 \).

Step-by-Step Solution:

1. Calculate \(\frac{\partial P}{\partial y}\):
2. Calculate \(\frac{\partial Q}{\partial x}\):
3. Set the expressions \(\frac{\partial P}{\partial y}\) and \(\frac{\partial Q}{\partial x}\) equal for the field to be conservative:
4. Equate coefficients of like terms:
   • Comparing the coefficients of \(y\), we get \(2a = 4 \Rightarrow a = 2\).
   • Comparing the coefficients of \(x\), we get \(b = 8\).
5. Thus, the values of \(a\) and \(b\) for which the force field is conservative are \(a = 2\) and \(b = 8\).

Conclusion: The correct answer is \(a = 2\) and \(b = 8\), which is supported by the examination of partial derivatives to confirm conservativeness of the force field.