A battery with voltage \(V\) sends a current \(I\) through a resistor \(R\). If \(R\) is doubled, \(I\) (A) stays the same. (B) quadruples. (C) halves. (D) doubles. (E) goes down by a factor of four.

Imagine electric current as a stream of electrons moving through a wire. Just like water flowing through a pipe, electric current refers to this flow of charge, usually measured in amperes (A). Current is not just a random movement but a directed flow from positive to negative potential. This is a fundamental concept in electrical circuits and is symbolized by the letter 'I' in equations.

Now, for a given voltage provided by, say a battery, the amount of current flowing through a circuit depends on the resistance it encounters. In our textbook example, the current 'I' from a battery going through a resistor 'R' is calculated using Ohm's Law. Changing the resistance, such as doubling 'R', influences this current, which brings us neatly to the relationship between resistance and current.

Electrical resistance can be likened to the friction experienced by water as it flows through a pipe; it hinders the smooth flow of electric current. Resistance, noted as 'R' in formulas, is measured in ohms (Ω) and is a property inherent to the materials used in the circuit. Different materials resist electric current to varying degrees - metals, for instance, generally have low resistance and allow current to flow easily. Conversely, non-metals tend to pose a greater resistance.

In the context of our textbook exercise, resistance plays a pivotal role. If 'R' is increased, naturally, 'I' decreases. This inverse relationship is at the heart of understanding electric circuits. Doubling the resistance doesn’t mean a double decrease in current; actually, as Ohm's Law displays, the current is halved (point C). This is essential when designing circuits or troubleshooting electrical issues, as correct resistance values are imperative to ensure the desired current flow.

Ohm's Law articulates the voltage-current relationship in a precise mathematical formula: \( V = IR \). This equation tells us that voltage (V) across a component in an electric circuit is the product of the current (I) flowing through it and the resistance (R) it presents.

When dealing with our equation and thinking about the resistance getting doubled, it's clear that if voltage remains constant and resistance goes up, current must go down. This sees a direct application in our textbook example, where we concluded that the current halves when the resistance doubles. It's critical to grasp that in a circuit, if you're knowledgeable about any two of these variables - voltage, current, and resistance - you can calculate the third. Understanding this interdependence is key to both solving circuit problems and to practical electrical design and analysis.