Draw E and V versus r on the same graph for a point charge.

Options

A. Parabola
B. Hyperbola
C. Straight line
D. Rectangular hyperbola

Answer

To draw the graphs of electric field \( E \) and electric potential \( V \) versus distance \( r \) for a point charge, we will follow these steps:

### Step 1: Understand the formulas
The electric field \( E \) due to a point charge \( Q \) at a distance \( r \) is given by:
\[ E = \frac{1}{4 \pi \epsilon_0} \frac{Q}{r^2} \]
The electric potential \( V \) due to a point charge \( Q \) at a distance \( r \) is given by:
\[ V = \frac{1}{4 \pi \epsilon_0} \frac{Q}{r} \]

### Step 2: Identify the behavior of \( E \) and \( V \)
- The electric field \( E \) varies inversely with the square of the distance \( r \) (i.e., \( E \propto \frac{1}{r^2} \)).
- The electric potential \( V \) varies inversely with the distance \( r \) (i.e., \( V \propto \frac{1}{r} \)).

### Step 3: Analyze the graphs
- The graph of \( E \) versus \( r \) will be a rectangular hyperbola that approaches the axes as \( r \) increases, indicating that \( E \) decreases rapidly as \( r \) increases.
- The graph of \( V \) versus \( r \) will also be a rectangular hyperbola but will decrease more slowly compared to \( E \).

### Step 4: Draw the axes
- On the x-axis, plot the distance \( r \).
- On the y-axis, plot both \( E \) and \( V \) (you can use different colors or line styles to distinguish them).

### Step 5: Sketch the graphs
- For \( E \), draw a curve that starts high when \( r \) is small and approaches zero as \( r \) increases, following the \( \frac{1}{r^2} \ behavior. - For \( V \), draw a curve that also starts high when \( r \) is small but decreases more gradually than \( E \), following the \( \frac{1}{r} \) behavior.

### Step 6: Label the graph
- Clearly label the curves for \( E \) and \( V \).
- Indicate the axes with appropriate units (e.g., \( E \) in N/C, \( V \) in volts, and \( r \) in meters).

### Final Graph
The final graph will show two hyperbolic curves: one for \( E \) that decreases sharply and one for \( V \) that decreases more gradually.

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