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Let's break down and solve the problems presented in the image.

### Problem 1:
The first question deals with the properties of an electrostatic equilibrium:

1. **When the electric potential at some point on the metal's surface has the value \( V \), the potential at a different point on the surface:**
    - For a conductor in electrostatic equilibrium, the electric potential is the same everywhere on the surface. Therefore, the correct answer is:
    - \(\text{(B) equals } V.\)

2. **When the electric field strength at some point on the body’s surface has the value \( E \), the field strength at a different point on the surface:**
    - In electrostatic equilibrium, the electric field on the surface of a conductor can vary depending on the shape of the conductor, but inside, the electric field is zero. Thus, the electric field on the surface can differ from point to point.
    - The correct answer is:
    - \(\text{(C) always differs from } E.\)

### Problem 2:
**Calculate the potential \( V \) at a point 5.97 m from the center of the charged piece of platinum.**

We can use the formula for the electric potential \( V \) due to a point charge, since the potential due to a spherically symmetric charge distribution behaves as though all the charge is concentrated at the center.

\[
V = \frac{k \cdot Q}{r}
\]

Where:
- \( k \) is Coulomb's constant, \( k = 8.99 \times 10^9 \, \text{N} \cdot \text{m}^2 / \text{C}^2 \).
- \( Q = -8.99 \times 10^{-9} \, \text{C} \) (charge of the platinum piece).
- \( r = 5.97 \, \text{m} \) (distance from the center).

Plugging in the values:

\[
V = \frac{(8.99 \times 10^9) \cdot (-8.99 \times 10^{-9})}{5.97}
\]

Let's calculate this value.The electric potential \( V \) at a point 5.97 m from the center of the charged piece of platinum is approximately \(-13.54 \, \text{V}\).

### Summary:
1. For the first part:
   - The potential \( V \) on the surface of the conductor is the same everywhere.
   - The electric field strength \( E \) can vary at different points on the surface.
   
2. For the second part:
   - The calculated potential \( V \) at 5.97 m from the center is \(-13.54 \, \text{V}\).

Would you like more details on any part of this solution, or do you have any questions? 

Here are 5 related questions to expand on this topic:

1. How does the shape of a conductor affect the distribution of charge on its surface?
2. What is the electric field inside a conductor in electrostatic equilibrium?
3. How would the potential change if the point was closer or further away from the charge?
4. Can we treat any charged object as a point charge for calculating potential? Why or why not?
5. How does the concept of equipotential surfaces apply to conductors?

**Tip:** Always ensure that units are consistent when calculating electric potential or field, especially when using formulas involving Coulomb's constant.

An isolated, irregularly shaped piece of platinum carries -8.99 x 10^-9 C of charge and is in electrostatic equilibrium. The size of this body is about 4 mm. 

When the electric potential at some point on the metal’s surface has the value V, the potential at a different point on the surface: 
A. may equal V
B. equals V
C. always differs from V

When the electric field strength at some point on the body’s surface has the value E, the field strength at a different point on the surface: 
A. may equal E
B. equals E
C. always differs from E

Calculate the potential V at a point 5.97 m from the center of the charged piece of platinum.

Understand the electrostatic equilibrium of a charged platinum body with -8.99 x 10^-9 C and calculate the electric potential at 5.97 m. Learn about electric fields and potentials on a conductor's surface.

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