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The problem shown involves charges \( q_2 \), \( q_6 \), and \( q_1 \), and it is asking for the change in potential energy \( \Delta U''' \) when charge \( q_1 \) is moved from point \( P \) to point \( R \). The charges \( q_2 \) and \( q_6 \) are both given as \( -5.50 \, \mu C \), and their relative distances from each other and from the origin are given. Let's break down the problem step by step.

### Key Details:
- Charge \( q_2 \) is at the origin.
- Charge \( q_6 \) is located at a distance \( d = d_1 + d_2 = 7.00 \, \text{cm} \) from the origin.
- Charge \( q_1 \) is being moved from point \( P \) (at a distance \( d_1 \) from \( q_2 \)) to point \( R \) (closer to \( q_2 \)).
- All charges involved have a magnitude of \( 5.50 \, \mu C \) and are negative.

### Formula for Potential Energy:
The potential energy \( U \) of a charge in the presence of other charges is given by:

\[
U = k_e \cdot \frac{q_1 q_2}{r}
\]

Where:
- \( k_e \) is Coulomb's constant \( (8.99 \times 10^9 \, \text{N} \cdot \text{m}^2 / \text{C}^2) \),
- \( q_1 \) and \( q_2 \) are the charges involved,
- \( r \) is the distance between the charges.

The change in potential energy \( \Delta U''' \) is the difference between the potential energy at point \( R \) and at point \( P \).

\[
\Delta U''' = U_R - U_P
\]

Let's calculate this. We need to determine the potential energies at both points \( R \) and \( P \), then subtract them.

### Step 1: Calculate Distances
From the diagram:
- Distance between \( q_1 \) at point \( P \) and \( q_2 \) is \( d_1 \).
- Distance between \( q_1 \) at point \( P \) and \( q_6 \) is \( d_2 \).
- Point \( R \) is closer to \( q_2 \), but we need to find its exact position from the diagram.

Once the distances are clear, we'll plug the values into the potential energy formula.

Would you like me to proceed with specific calculations, or clarify any of the distances/steps mentioned so far?

Charges q3 and q5 are now replaced by two charges, q2 and q6, that have equal magnitude and sign, -5.50 μC. Charge q2 is located at the origin and charge q6 is located a distance d = d1 + d2 = 7.00 cm from the origin, as shown. What is the change ΔU''' in potential energy if charge q1 is now moved from point P to point R?

This problem explores the change in potential energy (ΔU''') when a charge q1 is moved between points P and R in the presence of two charges, q2 and q6, both with a magnitude of -5.50 μC. Learn how to apply the formula for electric potential energy and calculate the difference between two points.

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