If potential energy is given by `U=(a)/(r^(2))-(b)/(r)`. Then find out maximum force. (given `a=2,b=4)

Options

A. `-(16)/(27)N`
B. `-(32)/(27)N`
C. `+(32)/(27)N`
D. `+(16)/(27)N`

Explanation

To solve the problem step by step, we start with the given potential energy function and derive the force from it.

### Step 1: Write down the potential energy function
The potential energy \( U \) is given by:
\[ U = \frac{a}{r^2} - \frac{b}{r} \]
where \( a = 2 \) and \( b = 4 \).

### Step 2: Find the force from potential energy
The force \( F \) in a potential field is given by:
\[ F = -\frac{dU}{dr} \]
We need to differentiate \( U \) with respect to \( r \).

### Step 3: Differentiate the potential energy function
Calculating the derivative:
\[ \frac{dU}{dr} = \frac{d}{dr}\left(\frac{a}{r^2}\right) - \frac{d}{dr}\left(\frac{b}{r}\right) \]
Using the power rule:
\[ \frac{d}{dr}\left(\frac{a}{r^2}\right) = -2\frac{a}{r^3} \quad \text{and} \quad \frac{d}{dr}\left(\frac{b}{r}\right) = -\frac{b}{r^2} \]
Thus,
\[ \frac{dU}{dr} = -2\frac{a}{r^3} + \frac{b}{r^2} \]

### Step 4: Substitute values of \( a \) and \( b \)
Substituting \( a = 2 \) and \( b = 4 \):
\[ \frac{dU}{dr} = -2\frac{2}{r^3} + \frac{4}{r^2} = -\frac{4}{r^3} + \frac{4}{r^2} \]

### Step 5: Write the expression for force
Now, substituting back into the force equation:
\[ F = -\left(-\frac{4}{r^3} + \frac{4}{r^2}\right) = \frac{4}{r^3} - \frac{4}{r^2} \]
This simplifies to:
\[ F = \frac{4}{r^3} - \frac{4}{r^2} \]

### Step 6: Find the maximum force
To find the maximum force, we need to set the derivative of \( F \) with respect to \( r \) to zero:
\[ \frac{dF}{dr} = -\frac{12}{r^4} + \frac{8}{r^3} \]
Setting this equal to zero:
\[ -\frac{12}{r^4} + \frac{8}{r^3} = 0 \]
Multiplying through by \( r^4 \) (assuming \( r \neq 0 \)):
\[ -12 + 8r = 0 \implies 8r = 12 \implies r = \frac{3}{2} \]

### Step 7: Substitute \( r \) back to find maximum force
Now substituting \( r = \frac{3}{2} \) back into the force equation:
\[ F_{\text{max}} = \frac{4}{\left(\frac{3}{2}\right)^3} - \frac{4}{\left(\frac{3}{2}\right)^2} \]
Calculating:
\[ F_{\text{max}} = \frac{4}{\frac{27}{8}} - \frac{4}{\frac{9}{4}} = \frac{32}{27} - \frac{16}{9} \]
Finding a common denominator (which is 27):
\[ F_{\text{max}} = \frac{32}{27} - \frac{48}{27} = -\frac{16}{27} \]

### Final Answer
Thus, the maximum force is:
\[ F_{\text{max}} = -\frac{16}{27} \]