A line chargeOf Royal is equal to 20 nano coulom per metre is uniformly distributed on circular ring of radius 15CM located at z is equals to zero assuming centre of ring at origin determined electric field intensity at 0,0 20cm CM

To determine the **electric field intensity** at point \( P(0, 0, 20 \text{ cm}) \) due to a uniformly distributed line charge on a circular ring, we will use the following approach.

### Step 1: Given Data
- **Charge density**: \( \lambda = 20 \, \text{nC/m} = 20 \times 10^{-9} \, \text{C/m} \)
- **Radius of the ring**: \( R = 15 \, \text{cm} = 0.15 \, \text{m} \)
- **Position of point**: \( P(0, 0, z_P) \), where \( z_P = 20 \, \text{cm} = 0.20 \, \text{m} \)
- The ring is located in the \( z = 0 \) plane and its center is at the origin.

### Step 2: Symmetry of the Problem
Due to the symmetry of the ring, the horizontal components of the electric field will cancel out, leaving only the **z-component** of the electric field.

### Step 3: Electric Field at a Point on the Axis of the Ring
The formula for the electric field intensity along the axis of a uniformly charged ring is:

\[
E_z = \frac{1}{4\pi \epsilon_0} \cdot \frac{z_P \lambda}{(R^2 + z_P^2)^{3/2}} \cdot 2\pi R
\]

Where:
- \( \epsilon_0 = 8.854 \times 10^{-12} \, \text{F/m} \) (permittivity of free space)
- \( z_P \) is the distance of the point from the plane of the ring along the \( z \)-axis
- \( \lambda \) is the linear charge density
- \( R \) is the radius of the ring

### Step 4: Substituting the Values
Substituting the given values into the formula:

\[
E_z = \frac{1}{4\pi \times 8.854 \times 10^{-12}} \cdot \frac{0.20 \times 20 \times 10^{-9}}{(0.15^2 + 0.20^2)^{3/2}} \cdot 2\pi \times 0.15
\]

We will now calculate the result step by step.

The electric field intensity at point \( P(0, 0, 20 \, \text{cm}) \) due to the uniformly charged circular ring is approximately **2168.51 N/C** in the positive \( z \)-direction.

Would you like further details or explanations?

Here are 5 related questions to explore further:

1. How does the electric field change if the charge density is doubled?
2. What is the electric field at the center of the ring?
3. How would the electric field change if the point is moved further along the \( z \)-axis?
4. Can we calculate the potential at the same point due to this charged ring?
5. How would the electric field change if the radius of the ring was increased?

**Tip:** In problems with high symmetry, such as rings or spheres, many components of the field cancel out due to symmetry, leaving only the field along certain axes.

A line charge of Royal is equal to 20 nano coulomb per meter is uniformly distributed on a circular ring of radius 15 cm located at z = 0. Assuming the center of the ring is at the origin, determine the electric field intensity at (0, 0, 20 cm).

Determine the electric field intensity at point (0, 0, 20 cm) caused by a uniformly distributed line charge of 20 nC/m on a circular ring with a radius of 15 cm, located at z = 0. This solution uses Coulomb's law and electric field symmetry. Suitable for undergraduate students studying electromagnetism.

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