What's the role for official intervention in the Balance of Payments? If the Balance of Payments (BP) is defined as 
$$BP=\text{Current Account}+\text{Capital Account}$$
where: 
$$\text{Current Account}=X-M+\text{Net Income Interest}$$ 
and 
$$\text{Capital Account}=\text{F}-\Delta \text{R}$$

Assume current account is positive, and $\text{Net Income Interest}=0$ for simplifying. Then there are 2 remaining ways for the balance of payments to be theoretically equal to zero:

1)$\text{F}&lt;0$ and $\Delta R=0$ i.e. by home country acquiring foreign assets, or 

2)  $\text{F}=0$ and $\Delta R&gt;0$ i.e. by the domestic Central Bank (CB) to increase their foreign exchange rate reserves.

Why would there be a need for official intervention by having the domestic central bank (CB) to increase its foreign reserves, if there was no buying of foreign assets ($\text{F}=0$), of ?

Any help would be appreciated.
The Balance of Payments is supposed to be a form of double-entry bookkeeping, in that every transaction has two opposite effects, often in different accounts.  Hence the word balance. In fact it might be considered quadruple-entry bookkeeping, as there are another two effects on the foreign side too.

For a country's Current Account to be in surplus there must be some net income, for example from the sales of exported goods and services exceeding the purchase of imported goods and services.

The surplus could be represented by a debt owed by the foreign importer to the domestic exporter. If the exporter uses this to invest in the foreign country then this counts in the Capital Account (or the Financial Account as the IMF calls it) which balances the Current Account.  But even if it simply remains as debt or as the exporter's balance in a foreign bank, this still counts as other investment and balances in the same way.  If the foreign importer pays in domestic currency then that reduces the outstanding liabilities to foreigners, again with the same net effect on the Financial Account. 

If somebody actually turns a debt into money, then it amounts to an investment in the liabilities of the foreign money-issuing authority (typically its Central Bank).  Bringing the foreign money home and selling it to somebody else domestically for domestic currency makes no difference to the impact on the Current or Financial Accounts, unless you sell foreign currency to the domestic Central Bank in which case it turns into a change in the Central Bank's Reserves.