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Let's break down and solve the two questions shown in the image.

### Question 3:
Three particles have charges of equal magnitude \( Q \), fixed at the corners of an equilateral triangle. Two charges are negative, and one is positive, with the goal of determining the direction of the resultant electric field at point \( P \), the center of the triangle.

1. **Charges Configuration**:
   - The two charges on the bottom corners are negative (\(-Q\)), and the top corner has a positive charge (\(+Q\)).
   
2. **Electric Field Contributions**:
   - The electric field due to a positive charge is directed **away** from the charge.
   - The electric field due to a negative charge is directed **towards** the charge.
   
   So, we need to determine the direction of the resultant electric field from all three charges:
   - The positive charge at the top of the triangle creates a downward electric field towards point \( P \).
   - The two negative charges on the bottom corners create electric fields pointing towards them, pulling \( P \) to the left and right, respectively. However, these horizontal components will cancel each other out, leaving only a vertical component pointing upwards.

3. **Resultant Field**:
   - The upward components from the two bottom charges will add together and cancel part of the downward component from the positive charge. But since the upward components are stronger, the resultant field will point upwards.
   
   **Answer**: The direction of the resultant electric field is **upward**, so the correct choice is **e**.

---

### Question 4:
Two isolated charged metal spheres \( X \) and \( Y \) are shown. Sphere \( X \) has a larger radius than sphere \( Y \), but the magnitudes of the electric fields just outside their surfaces are the same. The question asks how the charge on sphere \( X \) compares with the charge on sphere \( Y \).

1. **Electric Field Relation**:
   The electric field \( E \) outside a sphere is related to the charge \( Q \) on the sphere and its radius \( R \) by:
   \[
   E = \frac{kQ}{R^2}
   \]
   where \( k \) is Coulomb's constant.

2. **Equal Electric Fields**:
   Since the electric fields just outside the surfaces of both spheres are the same:
   \[
   \frac{kQ_X}{R_X^2} = \frac{kQ_Y}{R_Y^2}
   \]
   This simplifies to:
   \[
   \frac{Q_X}{R_X^2} = \frac{Q_Y}{R_Y^2}
   \]

3. **Larger Radius for \( X \)**:
   Given that \( R_X > R_Y \), the charge \( Q_X \) on sphere \( X \) must be **greater** than the charge \( Q_Y \) on sphere \( Y \) to maintain the same electric field.

   **Answer**: The charge on sphere \( X \) is greater, so the correct choice is **a**.

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Do you need further details on these solutions or have any questions? 

Here are 5 related questions to deepen your understanding:
1. How would the electric field at point \( P \) change if all charges were positive?
2. What would happen to the electric field at point \( P \) if the charges were not placed symmetrically?
3. How does the direction of an electric field change as you move farther away from the source charge?
4. How does the surface charge density of the two spheres compare given their electric fields are the same?
5. How would the force between two spheres change if the radii of the spheres were doubled while keeping the charges constant?

**Tip**: For problems involving electric fields and charges, always consider the symmetry of the charge distribution when simplifying the resultant field or force.

Three particles have charges of equal magnitude Q and are fixed at the corners of an equilateral triangle. Two of the charges are negative; the other is positive. Which of the following vectors best represents the direction of the resultant electric field at point P, the center of the triangle?
The two charged metal spheres X and Y are far apart, and each is isolated from all other charges. The radius of sphere X is greater than that of sphere Y, and the magnitudes of the electric fields just outside their surfaces are the same. How does the charge on sphere X compare with that on sphere Y?

Solve two physics problems involving electric fields. The first problem covers the direction of the resultant electric field at the center of an equilateral triangle with point charges. The second problem compares the charges of two isolated metal spheres with equal electric fields. Suitable for Grades 11-12 physics students.

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