Question

A Wheatstone bridge is constructed using a \(1.00-\mathrm{m}-\) long Nichrome wire (the purple line in the figure) with a conducting contact that can slide along the wire. A resistor, \(R_{1}=\) \(100 . \Omega\), is placed on one side of the bridge, and another resistor, \(R,\) of unknown resistance, is placed on the other side. The contact is moved along the Nichrome wire, and it is found that the ammeter reading is zero for \(L=25.0 \mathrm{~cm} .\) Knowing that the wire has a uniform cross section throughout its length, determine the unknown resistance.

Short answer

Answer: The unknown resistance (R) in the Wheatstone Bridge is 300 Ω.

Step-by-step solution

Write down the balanced condition of Wheatstone Bridge

In a balanced Wheatstone Bridge, the ratio of resistances across opposite sides of the bridge is equal. So, if we denote the two lengths of the Nichrome wire where the ammeter reads zero as \(L_{1}\) and \(L_{2}\), then we have: \(\frac{R_{1}}{R} = \frac{L_{1}}{L_{2}}\).

Substitute the given values

We are given that \(R_{1} = 100\,\Omega\) and the total length of the wire is \(1.00\,\mathrm{m}\). We are also told that the ammeter reads zero when the contact is at \(L = 25.0\,\mathrm{cm}\). So, \(L_{1} = 0.25\,\mathrm{m}\). Then, \(L_{2} = 1.00\,\mathrm{m} - 0.25\,\mathrm{m} = 0.75\,\mathrm{m}\).

Solve for the unknown resistance R

Now we can solve the balanced Wheatstone Bridge equation for the unknown resistance \(R\). Using the values of \(R_{1}\), \(L_{1}\), and \(L_{2}\), we get: \(\frac{100\,\Omega}{R} = \frac{0.25\,\mathrm{m}}{0.75\,\mathrm{m}}\) Rearranging the equation to solve for \(R\): \(R = 100\,\Omega \cdot \frac{0.75\,\mathrm{m}}{0.25\,\mathrm{m}}\)

Calculate the value of unknown resistance

Now we calculate the unknown resistance \(R\): \(R = 100\,\Omega \cdot 3\) \(R = 300\,\Omega\) So, the unknown resistance \(R\) in the Wheatstone Bridge is \(300\,\Omega\).

Key concepts

Electric Circuits

Electric circuits form the foundation of modern technology, allowing us to channel electrical energy in controlled ways to perform work, from lighting our homes to powering complex computer systems. What is fundamental to any electrical circuit is the path through which current flows. This path is created by connecting electrical components in various configurations, commonly in series or parallel.

In the example of the Wheatstone Bridge, a classic electric circuit, resistors are arranged in a diamond shape to precisely measure resistance. The primary advantage of this circuit is its capability to detect very small differences in resistance, which is why it's often used in sensor applications. The principle of a balanced Wheatstone Bridge—as seen in the exercise—is particularly important: when the bridge is balanced, the current through the ammeter is zero, implying no potential difference between the midpoints of the two resistor pairs. This technique, when wielded skilfully, can provide accurate measurements within electrical systems.

Resistance Calculations

Resistance calculations are vital in designing and understanding electric circuits. The resistance of an electrical component determines how much it resists the flow of electric current, similar to how a narrow pipe might restrict the flow of water. Ohm's Law, which states that the current through a conductor between two points is directly proportional to the voltage across the two points and inversely proportional to the resistance, is often used for these calculations.

In the context of our Wheatstone Bridge problem, calculations become straightforward when the bridge is balanced. The pivotal equation \( \frac{R_{1}}{R} = \frac{L_{1}}{L_{2}} \) is an expression of this balance. By carefully substituting known lengths from our Nichrome wire example and the known resistance, we can isolate and determine the unknown resistance. The simplification of this relationship into a neat, solvable equation highlights the elegance and power of resistance calculations in electrical engineering.

Nichrome Wire Properties

Nichrome, an alloy of nickel and chromium, is known for its high electrical resistance, durability, and ability to withstand high temperatures, making it an ideal material for heating elements and resistors in appliances like toasters and hair dryers. Nichrome wires exhibit a relatively stable resistance over a wide temperature range, which is why they are used in precision applications such as in the Wheatstone Bridge from our exercise.

For accurate experimental results, the uniform cross-section of the Nichrome wire ensures that the resistance per unit length remains consistent along the wire. This property allows us to infer that when we have a balanced Wheatstone Bridge, the ratio of the lengths of the wire segments directly correlates to the ratio of their resistances. This kind of predictability is paramount when conducting resistance calculations and designing circuits that rely on precision and repeatable outcomes.