A particle located in a one-dimensional potential field has its potential energy function as `U(x)(a)/(x^4)-(b)/(x^2)`, where a and b are positive constants. The position of equilibrium x corresponds to

Options

A. (a) `(b)/(2a)`
B. (b) `sqrt((2a)/(b))`
C. (c) `sqrt((2b)/(a))`
D. (d) `(a)/(2a)`

Explanation

To find the position of equilibrium \( x \) for the given potential energy function \( U(x) = \frac{a}{x^4} - \frac{b}{x^2} \), we need to follow these steps:

### Step 1: Understand the condition for equilibrium
At the position of equilibrium, the force acting on the particle must be zero. The force \( F(x) \) is related to the potential energy \( U(x) \) by the equation:
\[ F(x) = -\frac{dU}{dx} \]
Thus, for equilibrium:
\[ F(x) = 0 \implies -\frac{dU}{dx} = 0 \implies \frac{dU}{dx} = 0 \]

### Step 2: Differentiate the potential energy function
Now, we differentiate the potential energy function \( U(x) \):
\[ U(x) = \frac{a}{x^4} - \frac{b}{x^2} \]
Using the differentiation rule \( \frac{d}{dx}(x^n) = n x^{n-1} \), we find:
\[ \frac{dU}{dx} = \frac{d}{dx}\left(\frac{a}{x^4}\right) - \frac{d}{dx}\left(\frac{b}{x^2}\right) \]
Calculating each term:
\[ \frac{d}{dx}\left(\frac{a}{x^4}\right) = -4a x^{-5} = -\frac{4a}{x^5} \]
\[ \frac{d}{dx}\left(\frac{b}{x^2}\right) = -2b x^{-3} = -\frac{2b}{x^3} \]
Thus, we have:
\[ \frac{dU}{dx} = -\frac{4a}{x^5} + \frac{2b}{x^3} \]

### Step 3: Set the derivative to zero
Setting the derivative equal to zero for equilibrium:
\[ -\frac{4a}{x^5} + \frac{2b}{x^3} = 0 \]

### Step 4: Solve for \( x \)
Rearranging the equation gives:
\[ \frac{2b}{x^3} = \frac{4a}{x^5} \]
Multiplying both sides by \( x^5 \) to eliminate the denominators:
\[ 2b x^2 = 4a \]
Dividing both sides by 2:
\[ b x^2 = 2a \]
Now, solving for \( x^2 \):
\[ x^2 = \frac{2a}{b} \]
Taking the square root to find \( x \):
\[ x = \sqrt{\frac{2a}{b}} \]

### Step 5: Identify the position of equilibrium
Since \( a \) and \( b \) are positive constants, we only consider the positive root:
\[ x = \sqrt{\frac{2a}{b}} \]

### Final Answer
The position of equilibrium \( x \) corresponds to:
\[ x = \sqrt{\frac{2a}{b}} \]