## Mean Value Theorem (Classical Electrodynamics)

**Roel N. Baybayon**

MSPhysics1-MSU-IIT

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**Problem 1.10**

Prove the mean value theorem: For charge-free space the value of the electrostatic potential at any point is equal to the average of the potential over the surface of any sphere centered on that point.

**Proof:**

To prove this problem, we are going to use the **Green’s Second Identity** which is given by,

.

Choosing (*the scalar potential*), and * * be the integration variable, we have

. ** Eq.(1)**

Let us solve **Eq.(1)** term by term. For the first integral,

, since

, since if V contains .

For the second integral,

, since .

But because there is no charge in the volume that we are integrating(**Charge-free**) . So the second integral becomes

.

For the third integral,

.

For the fourth integral,

.

But , then

.

Using Divergence Theorem,

,

the fourth integral becomes

.

But and (again, this is true for a charge-free volume! ), then the fourth integral would be equal to zero, that is,

.

Thus, **Eq.(1)** is simplified into

.

Hence, the scalar potential is then equal to

. **Eq.(2)**

Now, we have proven the* mean value theorem*. **Eq.(2)** says that the potential at any point is equal to the average of the potential over the surface of any sphere centered on that point.

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