A particle located in one dimensional potential field has potential energy function`U(x)=(a)/(x^(2))-(b)/(x^(3))`, where a and b are positive constants. The position of equilibrium corresponds to x equal to

Options

A. `(3a)/(2b)`
B. `(2b)/(3a)`
C. `(2a)/(3b)`
D. `(3b)/(2a)`

Answer

To find the position of equilibrium for a particle in the given potential energy function \( U(x) = \frac{a}{x^2} - \frac{b}{x^3} \), we need to follow these steps:

### Step 1: Find the derivative of the potential energy function
To find the equilibrium position, we need to determine where the net force acting on the particle is zero. The force \( F \) is related to the potential energy \( U \) by the equation:
\[ F = -\frac{dU}{dx} \]
So, we first calculate the derivative of \( U(x) \):
\[ U(x) = \frac{a}{x^2} - \frac{b}{x^3} \]
Taking the derivative with respect to \( x \):
\[ \frac{dU}{dx} = -\frac{2a}{x^3} + \frac{3b}{x^4} \]

### Step 2: Set the derivative equal to zero
At equilibrium, the force is zero, which means:
\[ -\frac{dU}{dx} = 0 \]
Thus, we set the derivative equal to zero:
\[ -\left(-\frac{2a}{x^3} + \frac{3b}{x^4}\right) = 0 \]
This simplifies to:
\[ \frac{2a}{x^3} = \frac{3b}{x^4} \]

### Step 3: Rearrange the equation
To solve for \( x \), we can rearrange the equation:
\[ 2a \cdot x = 3b \]
This leads to:
\[ x = \frac{3b}{2a} \]

### Conclusion
The position of equilibrium corresponds to:
\[ x = \frac{3b}{2a} \]