To find the electric field at the point (0, 0, R) due to a quarter circular ring with a uniform charge distribution, we can follow these steps:
### Step 1: Understand the Geometry
The quarter circular ring lies in the first quadrant of the xy-plane with its center at the origin (0, 0, 0) and has a radius R. The point where we need to find the electric field is located at (0, 0, R).
### Step 2: Define the Charge Distribution
Let the total charge on the ring be Q. The charge is uniformly distributed along the quarter circular arc. The angle subtended by the quarter circle is π/2 radians.
### Step 3: Set Up the Infinitesimal Charge Element
Consider an infinitesimal charge element \( dQ \) on the ring. The charge can be expressed as:
\[
dQ = \frac{Q}{\frac{\pi}{2}} d\theta
\]
where \( d\theta \) is the infinitesimal angle in radians.
### Step 4: Calculate the Electric Field Contribution from \( dQ \)
The distance from the charge element \( dQ \) to the point (0, 0, R) is given by:
\[
r = \sqrt{R^2 + R^2} = R\sqrt{2}
\]
The electric field \( dE \) due to the charge \( dQ \) at the point (0, 0, R) is given by Coulomb's law:
\[
dE = k \frac{dQ}{r^2}
\]
Substituting for \( r \):
\[
dE = k \frac{dQ}{(R\sqrt{2})^2} = k \frac{dQ}{2R^2}
\]
### Step 5: Determine the Direction of \( dE \)
The electric field vector \( dE \) has components in the x, y, and z directions. The components can be expressed in terms of \( \theta \):
- The x-component: \( dE_x = dE \cdot \cos(\theta) \)
- The y-component: \( dE_y = dE \cdot \sin(\theta) \)
- The z-component remains unchanged: \( dE_z = dE \)
### Step 6: Integrate Over the Quarter Circle
The total electric field \( E \) at the point (0, 0, R) is the integral of \( dE \) over the angle from 0 to \( \frac{\pi}{2} \):
\[
E = \int_0^{\frac{\pi}{2}} dE
\]
Substituting for \( dE \):
\[
E = \int_0^{\frac{\pi}{2}} k \frac{Q}{2R^2} \frac{1}{2R^2} d\theta
\]
This integral will yield the total electric field components.
### Step 7: Calculate the Result
After performing the integration, we can find the components of the electric field and combine them to get the resultant electric field vector at the point (0, 0, R).
### Final Result
The electric field at the point (0, 0, R) due to the quarter circular ring can be expressed as:
\[
E = \frac{Q}{4\pi \epsilon_0 \sqrt{2} R^2} \left( -\hat{i} - \hat{j} + \hat{k} \right)
\]
