The problem involves calculating the magnitude of the electric field produced by a ring of charge. Let's break down the information provided:

• Linear charge density $\lambda = 170 \, \text{nC/m} = 170 \times 10^{-9} \, \text{C/m}$
• Radius of the ring $R = 81 \, \text{cm} = 0.81 \, \text{m}$
• Distance along the axis from the center of the ring to the point where the electric field is to be calculated, $z = 1.64 \, \text{m}$

Formula for the Electric Field on the Axis of a Ring of Charge:

The electric field $E$ along the axis of a ring of charge at a distance $z$ from the center is given by:

$E = \frac{1}{4 \pi \varepsilon_0} \cdot \frac{q z}{(z^2 + R^2)^{3/2}}$

Where:

• $q$ is the total charge on the ring
• $\varepsilon_0$ is the vacuum permittivity, $\varepsilon_0 = 8.854 \times 10^{-12} \, \text{C}^2/\text{N·m}^2$
• $z$ is the distance from the center of the ring along the axis
• $R$ is the radius of the ring

Steps to Solve:

1. Total Charge on the Ring: The total charge $q$ on the ring can be found by multiplying the linear charge density $\lambda$ by the circumference of the ring $2\pi R$: $q = \lambda \cdot 2\pi R$
2. Substitute Values into the Electric Field Formula: Once $q$ is calculated, substitute $q$, $R$, and $z$ into the electric field equation to get the result.

I will calculate this next.The magnitude of the electric field at a distance of $1.64 \, \text{m}$ from the center of the ring along its axis is approximately $2083.9 \, \text{N/C}$.

Would you like further details or have any questions?

Here are five related questions for deeper understanding:

1. How does the electric field change if the distance $z$ increases?
2. What happens to the electric field if the radius of the ring is doubled?
3. How would the electric field change if the charge density were halved?
4. What is the significance of the vacuum permittivity $\varepsilon_0$ in this formula?
5. How would the calculation change if the ring were not uniformly charged?

Tip: Remember that the electric field on the axis of a charged ring decreases with distance, and approaches zero as $z$ becomes much larger than $R$.