It seems like you're dealing with a circular ring of charge that has been cut so that 3/4 of it remains in the yz-plane. The ring's original radius is $R = 3.7 \, \text{cm}$ and the remaining charge distribution after the cut is $Q = 6.5 \times 10^{-12} \, \text{C}$. With this setup, you might be interested in calculating the electric field or potential at specific points or any other property related to this charge distribution.

Let's break this down:

Step 1: Total Charge Distribution

• The total charge on the full ring originally was $Q_{\text{total}}$, and the charge on the remaining segment after the cut is $Q = 6.5 \times 10^{-12} \, \text{C}$.

• Since the remaining segment is 3/4 of the original ring, the total charge on the original ring was:

  $Q_{\text{total}} = \frac{4}{3} Q = \frac{4}{3} \times 6.5 \times 10^{-12} \, \text{C} = 8.67 \times 10^{-12} \, \text{C}.$

Step 2: Charge Density

• The linear charge density $\lambda$ of the original ring can be calculated as:

  $\lambda = \frac{Q_{\text{total}}}{2\pi R} = \frac{8.67 \times 10^{-12} \, \text{C}}{2\pi \times 0.037 \, \text{m}} = 1.17 \times 10^{-10} \, \text{C/m}.$

• For the remaining 3/4 of the ring, the linear charge density $\lambda_{\text{segment}}$ will be the same since the charge distribution is uniform:

  $\lambda_{\text{segment}} = \frac{Q}{2\pi R \times \frac{3}{4}} = \frac{6.5 \times 10^{-12}}{2\pi \times 0.037 \times \frac{3}{4}} = 1.56 \times 10^{-10} \, \text{C/m}.$

This is the linear charge density for the remaining 3/4 of the ring.

Step 3: Electric Field Due to a Segment of Charge (if needed)

If you are looking to calculate the electric field at a particular point due to this remaining charge distribution, you would use the principle of superposition, integrating the contribution of each infinitesimal element of charge around the segment.

You would need more specific details (like the point where the field is to be calculated) to proceed with that calculation.

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Would you like me to continue with a specific calculation (e.g., the electric field or potential at a point)? Or do you have another direction in mind?