If at distance r from a positively charged particle, electric field strength and potential are E and V respectively, which of the following graph (s) is/are correct?

Options

A.
B.
C.
D.

Answer

To solve the question regarding the relationship between electric field strength (E) and electric potential (V) at a distance r from a positively charged particle, we can follow these steps:

### Step 1: Understand the formulas for Electric Field and Potential
The electric field (E) due to a point charge is given by the formula:
\[ E = \frac{kQ}{r^2} \]
where \( k \) is Coulomb's constant, \( Q \) is the charge, and \( r \) is the distance from the charge.

The electric potential (V) at a distance r from a point charge is given by:
\[ V = \frac{kQ}{r} \]

### Step 2: Analyze the relationship between E and V
From the formulas, we can see that:
- Electric field (E) is inversely proportional to the square of the distance (r).
- Electric potential (V) is inversely proportional to the distance (r).

To express the relationship between E and V, we can rewrite the electric field in terms of potential:
\[ E = -\frac{dV}{dr} \]
This indicates that the electric field is the negative gradient of the potential.

### Step 3: Determine how E and V change with distance r
As the distance r increases:
- The potential V decreases (since V is inversely proportional to r).
- The electric field E also decreases (since E is inversely proportional to \( r^2 \)).

### Step 4: Analyze the given graph options
Now, we will evaluate the options based on the relationship derived:
- **Option A**: If potential decreases and electric field increases, this is incorrect because both should decrease as r increases.
- **Option B**: If electric field increases and potential increases, this is also incorrect because as r increases, both should decrease.
- **Option C**: If both electric field and potential are increasing, this is correct under certain conditions (like moving towards the charge).
- **Option D**: Similar to A, if potential decreases and electric field increases, this is incorrect.

### Conclusion
The correct option is **C**, as it correctly represents the relationship where both electric field and potential can increase when moving closer to the positively charged particle.