Question

A wire is in the form of a tetrahedron shown in figure. The resistance of each wire is \(\mathrm{R}\). What is the resistance of the frame between the corners \(\mathrm{A}\) and \(\mathrm{B}\). (A) \((2 \mathrm{R} / 3)\) (B) \(2 \mathrm{R}\) (C) \(\mathrm{R}\) (D) \((\mathrm{R} / 2)\)

Short answer

The correct answer for the equivalent resistance between vertices A and B of the tetrahedron frame is (B) \(2R\).

Step-by-step solution

Label the vertices

Label the vertices of the tetrahedron frame as A, B, C, and D, where A and B are the corners we are interested in finding the resistance between.

Determine resistance combinations

The vertices form a total of 4 wire edges emanating from A (AB, AC, AD) and 3 from B (BC, BD, and BA, which is equal to AB). The resistances are in series with one another, and the resistances terminating at C and D are in parallel.

Express resistance of AC and AD in parallel

Since the resistance along AC and AD are R each and parallel, their equivalent resistance can be determined using the equation: \[R_{AC\_AD} =\frac{R \cdot R}{R + R} = \frac{R^2}{2R} = \frac{R}{2}\]

Express resistance of BC and BD in parallel

Similarly, the resistances between vertices B and C, and B and D, are in parallel, so we have: \[R_{BC\_BD} =\frac{R \cdot R}{R + R} = \frac{R^2}{2R} = \frac{R}{2}\]

Combine equivalent resistances in series

Now, we have the equivalent resistances (R, \(R_{AC\_AD}\), and \(R_{BC\_BD}\)) connected in series along the path from A to B. Therefore, their equivalent resistance is simply the sum of them: \[R_{AB} = R + \frac{R}{2} + \frac{R}{2} = R + R = 2R\]

Select the correct answer

The correct answer for the equivalent resistance between vertices A and B is (B) 2R.

Key concepts

Equivalent Resistance

When dealing with electrical circuits, especially complex ones like those in a tetrahedron shape, understanding equivalent resistance is crucial. Equivalent resistance simplifies a complex circuit into a simpler one with only a single resistor.
This makes calculations easier, especially when analyzing the circuit between two points. The concept of equivalent resistance allows us to combine multiple resistors into one "equivalent" resistor which has the same effect on the circuit as the original configuration.

• In series circuits, resistances simply add up.
  This is straightforward as the total resistance is the sum of individual resistances.
• In parallel circuits, the relationship is not additive.
  Instead, the equivalent resistance is found using the reciprocal formula: \[R_{eq} = \left( \frac{1}{R_1} + \frac{1}{R_2} + \ldots \right)^{-1}\]

By using these concepts together, complex circuits like those inside a tetrahedron can be simplified to a single resistance value.

Parallel Circuits

Parallel circuits are an arrangement where components are connected across common points or junctions, sharing the same voltage across all components. This is a distinct characteristic that differentiates them from series circuits. In a parallel circuit, each resistor is exposed to the same voltage.

• This results in an individual path for each current
• Thus, the total current is the sum of all individual currents through each path.

The main idea is that adding more resistors in parallel provides additional paths for the current to travel.
This effectively reduces the overall resistance.
The formula used to calculate the equivalent resistance in parallel is: \[R_{parallel} = \frac{1}{\left( \frac{1}{R_1} + \frac{1}{R_2} + \ldots \right)}\] In the tetrahedron example, the resistances corresponding to AC and AD, as well as BC and BD are calculated using this formula, remembering always to treat each set as its pathway.

Series Circuits

In simple terms, series circuits are designed in a manner where the components are connected end-to-end, forming a single pathway for current flow. This means the current flows through each component without branching. The defining feature of series circuits is the same current flows through each resistor.
This quality makes calculation straightforward.

• The total resistance in a series is the sum of all component resistances \[R_{series} = R_1 + R_2 + \ldots\]
• Voltage distribution occurs such that each component has a fraction of the total voltage.

In the tetrahedron problem, once the equivalent resistances from the parallel sections are found, these are combined in series along with other resistances.
This helps in deriving the total equivalent resistance between points A and B.

Electric Circuit Analysis

Electric circuit analysis is a foundational skill in electrical engineering, involving the examination and evaluation of electrical circuits to understand their behavior and characteristics.
It includes techniques to simplify circuits and calculate unknown values. Circuit analysis employs a variety of methods including:

• Ohm's Law
• Kirchhoff’s Laws
• Nodal and Mesh Analysis

These tools are applied to solve for currents, voltages, and resistances in any given circuit configuration.
In the context of the tetrahedron challenge, circuit analysis begins with labeling the circuit components. Properly identifying series and parallel sections is crucial to progressing toward a solution.
The completion of circuit analysis in this problem involves summing the resistances in series after having found the equivalent resistances from parallel sections. This strategic combination provides a precise assessment of the resistance between the points A and B.
Utilizing systematic analysis ensures accuracy and efficiency when dealing with complex circuits.