The energy density u is plotted against the distance r from the centre of a spherical charge distribution on a log-log scale. Find the magnitude of slope of obtained straight line.

Explanation

To solve the problem, we need to find the slope of the line when the energy density \( u \) is plotted against the distance \( r \) from the center of a spherical charge distribution on a log-log scale.

### Step-by-Step Solution:

1. **Understand the Energy Density Formula**:
The energy density \( u \) in an electric field is given by:
\[ u = \frac{1}{2} \epsilon_0 E^2 \]
where \( \epsilon_0 \) is the permittivity of free space and \( E \) is the electric field.

2. **Electric Field for a Spherical Charge Distribution**:
For a spherical charge distribution with total charge \( Q \), the electric field \( E \) at a distance \( r \) from the center (for \( r \) greater than the radius of the sphere) is given by:
\[ E = \frac{kQ}{r^2} \]
where \( k \) is Coulomb's constant.

3. **Substituting the Electric Field into the Energy Density Formula**:
Substitute the expression for \( E \) into the energy density formula:
\[ u = \frac{1}{2} \epsilon_0 \left(\frac{kQ}{r^2}\right)^2 \]
Simplifying this gives:
\[ u = \frac{1}{2} \epsilon_0 \frac{k^2 Q^2}{r^4} \]
This can be rewritten as:
\[ u = C \cdot \frac{1}{r^4} \]
where \( C = \frac{1}{2} \epsilon_0 k^2 Q^2 \) is a constant.

4. **Taking the Logarithm of Both Sides**:
To analyze the relationship on a log-log scale, take the logarithm of both sides:
\[ \log u = \log C - 4 \log r \]

5. **Identifying the Linear Relationship**:
The equation can be rearranged to resemble the linear form \( y = mx + b \):
\[ \log u = -4 \log r + \log C \]
Here, \( y = \log u \), \( x = \log r \), and the slope \( m = -4 \).

6. **Finding the Magnitude of the Slope**:
The magnitude of the slope is simply the absolute value of \( -4 \):
\[ \text{Magnitude of the slope} = 4 \]

### Final Answer:
The magnitude of the slope of the obtained straight line is \( 4 \).