A wire of resistance R is bent into a triangular pyramid as shown in figure with each segment having same length. The resistance between points A and B is R/n. The value of n is :

### Related Formula For a wire of total resistance R$R$ divided into N$N$ equal segments, the resistance of each segment r$r$ is: r = fracRN$$r = \frac{R}{N}$$ For a balanced Wheatstone bridge with resistors of resistance r$r$, the central arm can be neglected because no current flows through it. ### Core Logic The triangular pyramid has 6$6$ segments of equal length. Since the total resistance of the wire is R$R$: r = fracR6$$r = \frac{R}{6}$$ Let the four vertices of the pyramid be A$A$, B$B$, C$C$, and D$D$. Terminals are at A$A$ and B$B$. The segments are: - AB$AB$ (direct path between terminals, resistance r$r$) - AC$AC$, BC$BC$, AD$AD$, BD$BD$ (forming a closed quadrilateral network between A$A$ and B$B$ with bridge arm CD$CD$) - CD$CD$ (bridge arm connecting the midpoints, resistance r$r$) ### Step 1: Simplify the Network By symmetry, the potentials at C$C$ and D$D$ are equal when a voltage is applied across A$A$ and B$B$. Thus, the bridge is balanced, and no current flows through the segment CD$CD$. We can remove segment CD$CD$ from the calculations: - The path A to C to B$A \to C \to B$ consists of two resistors in series: r + r = 2r$r + r = 2r$. - The path A to D to B$A \to D \to B$ also consists of two resistors in series: r + r = 2r$r + r = 2r$. - The direct path A to B$A \to B$ has a single resistor r$r$. These three parallel branches are connected between A$A$ and B$B$. ### Step 2: Calculate Equivalent Resistance The equivalent resistance R_AB$R_{AB}$ is: frac1R_AB = frac12r + frac12r + frac1r = frac1r + frac1r = frac2r$$\frac{1}{R_{AB}} = \frac{1}{2r} + \frac{1}{2r} + \frac{1}{r} = \frac{1}{r} + \frac{1}{r} = \frac{2}{r}$$ R_AB = fracr2$$R_{AB} = \frac{r}{2}$$ Substitute r = fracR6$r = \frac{R}{6}$: R_AB = fracR/62 = fracR12$$R_{AB} = \frac{R/6}{2} = \frac{R}{12}$$ Comparing with R_AB = fracRn$R_{AB} = \frac{R}{n}$, we find n = 12$n = 12$. ### Pattern Recognition Sees: Resistor network formed by a 3D pyramid (6 identical edges, 4 nodes). Shortcut: A 6-resistor regular tetrahedron has an equivalent resistance of r/2$r/2$ across any two vertices. Since the total wire resistance is R$R$ and it's cut into 6 pieces, r = R/6 implies R_texteq = R/12$r = R/6 \implies R_{\text{eq}} = R/12$. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 12 Physics: Current Electricity