Symmetric 3D Resistor Networks: Applying path symmetry, axis-of-symmetry, and potential-division to evaluate the equivalent resistance of a cube of resistors across its body diagonals, face diagonals, or edges.

1. What is a Symmetric Resistor Network?

A symmetric resistor network is a network in which the arrangement of resistors is identical about one or more axes, planes or points of symmetry.

Because of symmetry, some points of the circuit must have the same electric potential.

Same symmetry → Same potential

This idea is extremely useful for solving a resistor cube. Instead of combining every resistor one by one, we first identify points having equal potential and then simplify the circuit.

2. Main Principle of Symmetry

Suppose two points A and B are completely equivalent by symmetry.

V_A = V_B

Therefore, the potential difference between them is:

V_A - V_B = 0

If a resistor R is connected between A and B:

I = (V_A-V_B)/R = 0

No current flows through a resistor connected between two equipotential points.

3. Resistor Cube

Consider a cube made of 12 identical resistors, each having resistance R.

The most important cases are:

Connection Across	Equivalent Resistance
Body Diagonal	R/3
Face Diagonal	3R/4
Edge	7R/12

These three results are important for competitive examinations and are frequently used as standard symmetry problems.

4. Cube Across Body Diagonal

Suppose current enters at one vertex A and leaves through the opposite vertex G.

By symmetry, the three resistors connected to A carry equal currents.

I/3

The three vertices adjacent to A are at the same potential. Therefore they form one equipotential group.

Similarly, the three vertices adjacent to G form another equipotential group.

The cube can therefore be reduced to three sections.

Section	Number of Resistors	Equivalent
First section	3 parallel	R/3
Middle section	6 parallel	R/6
Last section	3 parallel	R/3

These three equivalent resistances are in series:

R_eq = R/3 + R/6 + R/3

Therefore:

R_eq = 5R/6

Correction: The above grouping is not the correct grouping for the standard 12-edge cube across opposite vertices. The six middle edges do not all form a simple six-parallel group in the required topology. The correct symmetry result is:

R_body = 5R/6

5. Cube Across Face Diagonal

Now connect the source between two opposite vertices of one face.

The plane passing through the two terminals and the cube's symmetry axis allows several vertices to be identified as equipotential.

Using symmetry and reducing the resulting branches gives:

R_{face diagonal} = 3R/4

For a cube of twelve equal resistors R:

Across a face diagonal:

R_eq = 3R/4

6. Cube Across One Edge

Now connect the battery across the two ends of one edge.

The remaining network can be simplified by using planes of symmetry.

The standard result is:

R_edge = 7R/12

For a cube made of identical resistors:

R_edge = 7R/12

7. Important Cube Results

Terminal Arrangement	Equivalent Resistance
Body Diagonal	5R/6
Face Diagonal	3R/4
Edge	7R/12

Memory Trick: For the standard resistor cube:

Body → 5R/6
Face → 3R/4
Edge → 7R/12

8. Potential Division Method

The potential division method is useful after symmetry has grouped several nodes at equal potentials.

For resistors in series:

V₁/V₂ = R₁/R₂

For example, if two resistors R and 2R are in series across voltage V:

V_R = V × R/(R+2R)

Therefore:

V_R = V/3

and:

V_2R = 2V/3

In a symmetric cube, potential division helps determine the potentials of intermediate nodes and hence identify branches carrying zero current.

9. Path Symmetry

If several paths from the input terminal to the output terminal are identical, the current divides equally among those paths.

I_each = I/n

where n is the number of identical paths.

For example, if three identical resistors leave a node symmetrically, each carries one-third of the total current.

10. Axis and Plane of Symmetry

A cube has several symmetry axes and planes.

When the terminals are placed symmetrically, the corresponding vertices must have equal potentials.

V_A = V_B

A resistor joining such points carries no current.

I = 0

Therefore, that resistor may be removed for equivalent resistance calculation.

11. Step-by-Step Method

Step 1 — Identify the terminals

Mark the two points between which equivalent resistance is required.

Step 2 — Look for symmetry

Check whether the network is symmetric about an axis or plane.

Step 3 — Find equipotential points

V₁ = V₂

Step 4 — Remove zero-current resistors

I = 0

Step 5 — Combine series and parallel branches

R_parallel = (R₁R₂)/(R₁+R₂)

Step 6 — Find final equivalent resistance

R_eq = V/I

12. Important Formula Sheet

Concept	Formula
Ohm's Law	V = IR
Current	I = V/R
Series Resistance	R = R₁+R₂+...
Two Parallel Resistors	R = R₁R₂/(R₁+R₂)
n Equal Parallel Resistors	Req = R/n
Cube — Body Diagonal	5R/6
Cube — Face Diagonal	3R/4
Cube — Edge	7R/12

Section A — 1 Mark
10 MCQs

Q1. The equivalent resistance of a cube of 12 equal resistors R across a body diagonal is: 1 Mark

A) R/3

B) 3R/4

C) 5R/6

D) 7R/12

Correct Answer: C

Q2. The equivalent resistance of the same cube across a face diagonal is: 1 Mark

A) R/2

B) 3R/4

C) 5R/6

D) R

Correct Answer: B

Q3. The equivalent resistance across one edge of a resistor cube is: 1 Mark

A) R/3

B) R/2

C) 7R/12

D) 5R/6

Correct Answer: C

Q4. If two points have the same potential, the current through a resistor joining them is: 1 Mark

A) Maximum

B) Zero

C) Infinite

D) V/R²

Correct Answer: B

Q5. Symmetry in a resistor network is mainly used to find: 1 Mark

A) Equal potentials

B) Temperature

C) Capacitance

D) Magnetic flux

Correct Answer: A

Q6. Three identical resistors R in parallel have equivalent resistance: 1 Mark

A) 3R

B) R

C) R/3

D) 2R

Correct Answer: C

Q7. If identical paths carry the same current, the total current is: 1 Mark

A) I/n

B) nI

C) I²

D) I

Correct Answer: B

Q8. The resistance of two equal resistors R in parallel is: 1 Mark

A) 2R

B) R

C) R/2

D) R²

Correct Answer: C

Q9. A resistor carrying zero current has: 1 Mark

A) Zero potential difference

B) Infinite potential difference

C) Maximum current

D) Negative resistance

Correct Answer: A

Q10. The resistance of a resistor cube across a body diagonal is: 1 Mark

A) Greater than R

B) Equal to R

C) Less than R

D) Infinite

Correct Answer: C

Section B — 2 Marks
10 Conceptual + Numerical Questions

Q11. Three equal resistors of resistance R are connected in parallel. Find the equivalent resistance. 2 Marks

1/R_eq = 1/R+1/R+1/R = 3/R

R_eq = R/3

Q12. Why do equal-potential points occur in a symmetric network? 2 Marks

Because the network and terminal conditions are identical under the symmetry operation. Therefore, equivalent points must have the same potential.

Q13. A 6 Ω resistor and a 3 Ω resistor are connected in parallel. Find the equivalent resistance. 2 Marks

R_eq = (6×3)/(6+3)

R_eq = 2 Ω

Q14. What happens to the current through a resistor joining two equipotential points? 2 Marks

The current becomes zero because the potential difference across the resistor is zero.

Q15. Find the equivalent resistance of a cube across its body diagonal if each resistor is 6 Ω. 2 Marks

R_eq = 5R/6

= 5×6/6

R_eq = 5 Ω

Q16. Find the equivalent resistance across a face diagonal of a cube whose each resistor is 8 Ω. 2 Marks

R_eq = 3R/4

= 3×8/4

R_eq = 6 Ω

Q17. Find the equivalent resistance across one edge of a cube if each resistor is 12 Ω. 2 Marks

R_eq = 7R/12

= 7×12/12

R_eq = 7 Ω

Q18. State the three standard equivalent resistances of a resistor cube. 2 Marks

Body diagonal = 5R/6
Face diagonal = 3R/4
Edge = 7R/12

Q19. What is the purpose of a symmetry plane in a resistor network? 2 Marks

It helps identify corresponding points having equal potentials and allows zero-current branches to be removed.

Q20. If four equal resistors R are connected in parallel, find the equivalent resistance. 2 Marks

R_eq = R/4

R_eq = R/4

Section C — 3 Marks
10 CBSE + Competitive Questions

Q21. Derive the equivalent resistance of a resistor cube across a body diagonal. 3 Marks

Let each resistor be R. By symmetry, the three vertices adjacent to the first terminal are at the same potential. Therefore the three resistors connected to the input are in parallel:

R₁ = R/3

Similarly, the three resistors connected to the output terminal are:

R₃ = R/3

The six central resistors form the middle section:

R₂ = R/6

Hence:

R_eq = R/3 + R/6 + R/3

Therefore:

R_eq = 5R/6

Q22. A cube contains 12 equal resistors of 10 Ω each. Find its equivalent resistance across a body diagonal. 3 Marks

For body diagonal:

R_eq = 5R/6

Therefore:

R_eq = 5×10/6

R_eq = 25/3 Ω ≈ 8.33 Ω

Q23. A resistor cube has each edge resistance 3 Ω. Find the resistance across a face diagonal. 3 Marks

For face diagonal:

R_eq = 3R/4

Therefore:

R_eq = 3×3/4

R_eq = 9/4 Ω = 2.25 Ω

Q24. A cube is made of 24 Ω resistors. Find its equivalent resistance across one edge. 3 Marks

For an edge connection:

R_eq = 7R/12

Therefore:

R_eq = 7×24/12

R_eq = 14 Ω

Q25. Explain why the three branches connected to a vertex of a symmetric cube carry equal current when current enters that vertex. 3 Marks

The three branches are geometrically identical. The rest of the cube seen from each branch is also identical because of symmetry. Therefore, the three branches have the same resistance and the same potential difference. Hence their currents are equal:

I₁=I₂=I₃

If total current is I:

I₁=I₂=I₃=I/3

Q26. Explain how a zero-current branch can be removed from a symmetric resistor network. 3 Marks

Suppose two points A and B are equivalent by symmetry. Then:

V_A=V_B

Potential difference across the resistor joining them:

V_AB=0

Hence:

I=V_AB/R=0

Therefore the branch carries no current and can be removed without changing the equivalent resistance.

Q27. A cube has each resistor 18 Ω. Compare its equivalent resistance when connected across body diagonal and one edge. 3 Marks

Body diagonal:

R_body=5R/6=5×18/6=15 Ω

Edge:

R_edge=7R/12=7×18/12=10.5 Ω

Therefore:

R_body = 15 Ω
R_edge = 10.5 Ω

Q28. A potential difference of 12 V is applied across a cube of equal resistors 6 Ω along a body diagonal. Find the total current. 3 Marks

Equivalent resistance:

R_eq=5R/6

R_eq=5×6/6=5 Ω

Using Ohm's law:

I=V/R

I=12/5

I = 2.4 A

Q29. Explain the difference between body diagonal, face diagonal and edge connection of a resistor cube. 3 Marks

Edge: The two terminals are adjacent vertices connected directly by one edge.

Face diagonal: The terminals are opposite vertices on the same square face.

Body diagonal: The terminals are opposite vertices of the cube. Their equivalent resistances are different because the symmetry of the network relative to the terminals is different.

R_edge=7R/12

R_face=3R/4

R_body=5R/6

Q30. Describe the complete procedure for solving a symmetric 3D resistor network. 3 Marks

Step 1: Identify the two terminals.

Step 2: Look for axes, planes and paths of symmetry.

Step 3: Identify points having equal potentials.

Step 4: Remove resistors joining equipotential points because they carry zero current.

Step 5: Combine the remaining resistors using series and parallel rules.

Step 6: Use potential division wherever necessary.

Step 7: Finally calculate:

R_eq=V/I

The key idea is: symmetry first, reduction second.

Quick Revision Formula Sheet

V = IR

R_series=R₁+R₂+...

1/R_parallel=1/R₁+1/R₂+...

R_eq=V/I

Standard Resistor Cube

Body Diagonal → 5R/6

Face Diagonal → 3R/4

Edge → 7R/12

Competitive Exam Tip: Do not start combining resistors randomly. First mark the symmetry. If two points have equal potential, the resistor between them carries zero current. This is the most important trick in 3D resistor-cube problems.

Symmetric 3D Resistor Networks
CBSE + Competitive Physics Practice | 30 Questions with Answers & Solutions