In electrical circuits, combining voltage sources can simplify analysis, particularly when they are connected in series. When voltage sources are in series, their voltages can be summed up. For instance, if you have a 10-volt battery and a 5-volt battery connected in series, the total voltage can be calculated as:

$$ V_{\text{total}} = V_1 + V_2 $$

In this case, the combined voltage would be 15 volts. It's important to note that if the voltage sources are oriented in opposite directions, their voltages will subtract rather than add. For example, if a 10-volt source is pushing current in one direction and a 5-volt source is pushing in the opposite direction, the effective voltage can be determined by:

$$ V_{\text{effective}} = V_1 - V_2 $$

Here, the effective voltage would be 5 volts in the direction of the stronger source.

When analyzing the circuit, the direction of current flow can be determined by the orientation of the voltage sources. If both sources are pushing current in the same direction, the current will flow clockwise. Conversely, if they oppose each other, the stronger source dictates the overall direction of current flow.

To find the magnitude of the current in a circuit, Ohm's Law is applied, which states:

$$ V = I \cdot R $$

Rearranging this gives:

$$ I = \frac{V}{R} $$

For example, with a total voltage of 15 volts and a resistance of 5 ohms, the current can be calculated as:

$$ I = \frac{15 \text{ volts}}{5 \text{ ohms}} = 3 \text{ amps} $$

In the case where the effective voltage is 5 volts against a 5-ohm resistor, the current would be:

$$ I = \frac{5 \text{ volts}}{5 \text{ ohms}} = 1 \text{ amp} $$

This approach to merging voltages and calculating current is fundamental in circuit analysis, allowing for a clearer understanding of how components interact within an electrical system.