Two 10 cm diameter charged rings face each other, 30 cm apart. The left ring is charged to − 30 nC and the right ring is charged to + 30 nC.

What is the magnitude of the electric field E⃗ E→ at the midpoint between the two rings?

To find the magnitude of the electric field at the midpoint between two charged rings, we need to consider the contributions from both rings.

### Given:
- Diameter of each ring: 10 cm → Radius \( R = 5 \, \text{cm} = 0.05 \, \text{m} \)
- Distance between the rings: 30 cm → \( d = 0.30 \, \text{m} \)
- Left ring charge: \( Q_1 = -30 \, \text{nC} = -30 \times 10^{-9} \, \text{C} \)
- Right ring charge: \( Q_2 = +30 \, \text{nC} = +30 \times 10^{-9} \, \text{C} \)
- Midpoint is 15 cm from each ring.

### Approach:
We can use the symmetry and properties of electric fields created by charged rings.

The formula for the electric field along the axis of a ring of charge at a distance \( z \) from the center is:
\[
E = \frac{k \cdot Q \cdot z}{(z^2 + R^2)^{3/2}}
\]
Where:
- \( k = 8.99 \times 10^9 \, \text{N·m}^2/\text{C}^2 \) is Coulomb's constant,
- \( Q \) is the charge on the ring,
- \( z \) is the distance from the center of the ring to the point where the field is measured,
- \( R \) is the radius of the ring.

### Electric Field at the Midpoint:
At the midpoint between the rings, the electric fields from the two rings will point in opposite directions, so we sum their magnitudes:

- Distance to the midpoint from each ring is \( z = 0.15 \, \text{m} \).

The electric field from the left ring (negatively charged) at the midpoint will point towards the left ring, and the electric field from the right ring (positively charged) will point towards the right ring.

Since the charges are equal in magnitude but opposite in sign, the magnitudes of the fields from both rings will be the same, but they will add because they are in the same direction (toward the right due to the symmetry).

Thus, the net electric field at the midpoint is twice the electric field produced by one ring.

\[
E_{\text{net}} = 2 \cdot \frac{k \cdot Q_1 \cdot z}{(z^2 + R^2)^{3/2}}
\]

### Substituting values:
\[
E_{\text{net}} = 2 \cdot \frac{(8.99 \times 10^9) \cdot (30 \times 10^{-9}) \cdot 0.15}{(0.15^2 + 0.05^2)^{3/2}}
\]

\[
E_{\text{net}} = 2 \cdot \frac{(8.99 \times 10^9) \cdot (30 \times 10^{-9}) \cdot 0.15}{(0.0225 + 0.0025)^{3/2}}
\]

\[
E_{\text{net}} = 2 \cdot \frac{(8.99 \times 10^9) \cdot (30 \times 10^{-9}) \cdot 0.15}{0.025^{3/2}}
\]

\[
E_{\text{net}} = 2 \cdot \frac{(8.99 \times 10^9) \cdot (30 \times 10^{-9}) \cdot 0.15}{0.003953}
\]

\[
E_{\text{net}} = 2 \cdot \frac{4.0455 \times 10^{-7}}{0.003953}
\]

\[
E_{\text{net}} \approx 2 \cdot 102.33 \, \text{N/C}
\]

\[
E_{\text{net}} \approx 204.66 \, \text{N/C}
\]

### Final Answer:
The magnitude of the electric field at the midpoint is approximately **204.66 N/C**.

---

Would you like to explore this concept further or have any additional questions? 

Here are some related questions you might find interesting:
1. How does the electric field change if the distance between the rings is increased?
2. What happens if the charges on both rings are made equal in magnitude but both positive?
3. How would the electric potential at the midpoint differ from the electric field?
4. Can this method be extended to find the field at any point along the axis between the rings?
5. What if one of the rings is grounded? How would the electric field change?

**Tip:** Remember that electric fields from multiple sources can add or subtract depending on their directions!

Two 10 cm diameter charged rings face each other, 30 cm apart. The left ring is charged to −30 nC and the right ring is charged to +30 nC. What is the magnitude of the electric field E⃗ at the midpoint between the two rings?

Learn how to calculate the electric field at the midpoint between two oppositely charged rings. This solution covers concepts from electrostatics, including Coulomb's law and the superposition principle. The problem involves calculating the net electric field using the electric field formula for a ring. Suitable for advanced high school or undergraduate students.

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