 Points P , Q and R are located in an electric field set up by a set of charges that are fixed in three-dimensional space. 

A test charge of 5 C is being transferred from point P to point R via point Q. During the transfer from point P to point Q,  10 J of work is done by the electric field.  During the transfer from point Q to point R, 20 J of work is done against the electric field.

VPQ = electric potential difference between point P and point Q   

VQR = electric potential difference between point Q and point R

VRP = electric potential difference between point R and point P.

The above three potential differences are written using the double subscript notation.

Question 1 Answer

a.
VPQ = -2 V and VQR = -4 V and VRP = 2 V


b.
VPQ = 2 V and VQR = -4 V and VRP = -2 V


c.
VPQ = 2 V and VQR = -4 V and VRP = 2 V


d.
All of the other statements are incorrect

To solve this problem, we'll use the relationship between the work done by the electric field, the electric potential difference, and the charge being moved.

The electric potential difference \( V_{xy} \) between two points \( x \) and \( y \) in an electric field is related to the work \( W_{xy} \) done by the electric field to move a charge \( q \) from \( x \) to \( y \) by the equation:

\[
V_{xy} = -\frac{W_{xy}}{q}
\]

Let's calculate the potential differences:

1. **\( V_{PQ} \)**:
   - Work done by the electric field \( W_{PQ} = 10 \) J.
   - Charge \( q = 5 \) C.
   \[
   V_{PQ} = -\frac{10\ \text{J}}{5\ \text{C}} = -2\ \text{V}
   \]

2. **\( V_{QR} \)**:
   - Work done against the electric field \( W_{QR} = 20 \) J.
   - Work done against the electric field means the electric field would do negative work, so \( W_{QR} = -20 \) J.
   \[
   V_{QR} = -\frac{-20\ \text{J}}{5\ \text{C}} = 4\ \text{V}
   \]

3. **\( V_{RP} \)**:
   - We can calculate \( V_{RP} \) by using the fact that:
   \[
   V_{RP} = V_{RQ} + V_{QP} = -V_{PQ} - V_{QR}
   \]
   \[
   V_{RP} = -(-2\ \text{V}) - 4\ \text{V} = 2\ \text{V} + (-4\ \text{V}) = -2\ \text{V}
   \]

Given the options:

- **Option a:** \( V_{PQ} = -2\ \text{V} \), \( V_{QR} = -4\ \text{V} \), \( V_{RP} = 2\ \text{V} \) (Incorrect).
- **Option b:** \( V_{PQ} = 2\ \text{V} \), \( V_{QR} = -4\ \text{V} \), \( V_{RP} = -2\ \text{V} \) (Incorrect).
- **Option c:** \( V_{PQ} = 2\ \text{V} \), \( V_{QR} = -4\ \text{V} \), \( V_{RP} = 2\ \text{V} \) (Incorrect).
- **Option d:** All of the other statements are incorrect (Correct).

**Answer:** d. All of the other statements are incorrect.

Would you like any further details or explanations?

Here are 8 related questions you might find interesting:
1. What is the relationship between electric potential and electric potential energy?
2. How is electric potential difference related to the work done in moving a charge?
3. What happens to the electric potential when a positive charge moves in the direction of the electric field?
4. How does the sign of the work done relate to the direction of the electric field?
5. What is the significance of a positive versus a negative electric potential difference?
6. How can the concept of electric potential be extended to three-dimensional fields?
7. What are equipotential surfaces, and how do they relate to the electric field?
8. How does the principle of superposition apply to electric potential?

**Tip:** Always pay attention to the direction of the electric field and the sign of the work done when calculating potential differences.

Explore the relationship between electric potential, work done by an electric field, and potential difference in a complex electric field setup. Solve a problem involving charges at points P, Q, and R with detailed step-by-step calculations and explanations.

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