### Related Formula The torque vectau$\vec{\tau}$ experienced by a magnetic dipole in a uniform magnetic field vecB$\vec{B}$ is given by: vectau = vecM times vecB Rightarrow tau = MB sintheta$\vec{\tau} = \vec{M} \times \vec{B} \Rightarrow \tau = MB \sin\theta$ The potential energy U$U$ of the dipole is given by: U = -vecM cdot vecB = -MB costheta$U = -\vec{M} \cdot \vec{B} = -MB \cos\theta$ ### Core Logic Given parameters: - Torque tau = 80sqrt3mathrm~N~m$\tau = 80\sqrt{3}\mathrm{~N~m}$ - Angle theta = 60^circ$\theta = 60^{\circ}$ ### Step 1: Calculate the value of MB$MB$ Substitute the given values into the torque formula: 80sqrt3 = MB sin(60^circ)$80\sqrt{3} = MB \sin(60^{\circ})$ 80sqrt3 = MB left(fracsqrt32right)$80\sqrt{3} = MB \left(\frac{\sqrt{3}}{2}\right)$ MB = 160mathrm~J$MB = 160\mathrm{~J}$ ### Step 2: Calculate Potential Energy Using the potential energy expression: U = -MB cos(60^circ)$U = -MB \cos(60^{\circ})$ U = -160 times frac12 = -80mathrm~J$U = -160 \times \frac{1}{2} = -80\mathrm{~J}$ ### Pattern Recognition A standard dipole problem checking the relation between torque and potential energy. Note that torque goes with the sine of the angle, whereas potential energy goes with the negative cosine. Dividing torque by potential energy gives -tantheta$-\tan\theta$. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 12 Physics: Magnetism and Matter

### Related Formula The magnetic field inside a filled solenoid is: B_textnew = mu n I$B_{\text{new}} = \mu n I$ The magnetic field in air is: B_textold = mu_0 n I$B_{\text{old}} = \mu_0 n I$ Relative permeability mu_r$\mu_r$ and susceptibility chi$\chi$ are related by: mu_r = fracmumu_0 = 1 + chi$\mu_r = \frac{\mu}{\mu_0} = 1 + \chi$ ### Core Logic The percentage increase in magnetic field is: \% text change in B = fracB_textnew - B_textoldB_textold times 100\%$\% \text{ change in } B = \frac{B_{\text{new}} - B_{\text{old}}}{B_{\text{old}}} \times 100\%$ \% text change in B = fracmu - mu_0mu_0 times 100\% = (mu_r - 1) times 100\% = chi times 100\%$\% \text{ change in } B = \frac{\mu - \mu_0}{\mu_0} \times 100\% = (\mu_r - 1) \times 100\% = \chi \times 100\%$ ### Step 1: Substitute and Calculate Given susceptibility chi_mathrmmg = 1.2 times 10^-5$\chi_{\mathrm{mg}} = 1.2 \times 10^{-5}$: \% text change = 1.2 times 10^-5 times 100\% = 1.2 times 10^-3\%$\% \text{ change} = 1.2 \times 10^{-5} \times 100\% = 1.2 \times 10^{-3}\%$ Convert decimal to fraction: 1.2 times 10^-3\% = frac65 times 10^-3\%$1.2 \times 10^{-3}\% = \frac{6}{5} \times 10^{-3}\%$ ### Pattern Recognition Sees: Susceptibility and solenoid field percentage change. Shortcut: The fractional change in magnetic field when inserting a core is exactly equal to its magnetic susceptibility chi$\chi$. Thus, the percentage change is just chi times 100$\chi \times 100$. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 12 Physics: Magnetism and Matter

### Related Formula textMagnetic Flux: Phi = B cdot A implies textWeber$\text{Magnetic Flux: } \Phi = B \cdot A \implies \text{Weber}$ textMagnetic Intensity: H = fracBmu implies textAmpere/meter$\text{Magnetic Intensity: } H = \frac{B}{\mu} \implies \text{Ampere/meter}$ ### Core Logic Analyzing official units: - **(A) Magnetic induction** rightarrow$\rightarrow$ Gauss (CGS unit) rightarrow$\rightarrow$ **(III)** - **(B) Magnetic intensity** rightarrow$\rightarrow$ Ampere/meter rightarrow$\rightarrow$ (Note: List-II text erroneously says "Ampere meter" or missing solidus bar, actual standard is mathrmAcdot m^-1$\mathrm{A\cdot m^{-1}}$) rightarrow$\rightarrow$ **(IV)** - **(C) Magnetic flux** rightarrow$\rightarrow$ Weber rightarrow$\rightarrow$ **(II)** - **(D) Magnetic moment** rightarrow$\rightarrow$ textAmpere meter^2$\text{Ampere meter}^2$ rightarrow$\rightarrow$ **(I)** Due to typo variants in option strings on the primary list sheet, this question was technically **dropped by NTA**. If picking the closest appropriate intended layout configuration, option (2) presents the best approximation. ### Pattern Recognition Note that this question was officially declared dropped by NTA due to printing formatting discrepancies in the unit labels. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 12 Physics: Magnetism and Matter

### Related Formula The standard expression for the magnetic field B$B$ on the axial path at distance r$r$ from the magnetic center is given by : B propto frac1r^3$B \propto \frac{1}{r^3}$ Through logarithmic error differentiation: fracDelta BB = 3 cdot left(fracDelta rrright)$\frac{\Delta B}{B} = 3 \cdot \left(\frac{\Delta r}{r}\right)$ ### Core Logic Depending on how the evaluation tracks parameter variables, two interpretations arise: **Method 1 (Approximating without considering variations in ell$\ell$ independently) [cite: 708, 709]:** If the spatial uncertainty is tied entirely to the radial component variable r$r$ : fracDelta BB = 3 times 1\% = 3\%$\frac{\Delta B}{B} = 3 \times 1\% = 3\%$ **Method 2 (Accounting comprehensively for dimensional dependencies) [cite: 714, 715]:** If the error accumulation bounds combine tracking lengths alongside parameters: fracDelta BB = fracDelta ellell + 3left(fracDelta rrright) = 1\% + 3(1\%) = 4\%$\frac{\Delta B}{B} = \frac{\Delta \ell}{\ell} + 3\left(\frac{\Delta r}{r}\right) = 1\% + 3(1\%) = 4\%$ Both paths offer distinct structural insights depending on assumptions. The official valuation matrix accepts options reflecting both interpretations. ### Pattern Recognition In general engineering error evaluations, always look at power exponents. If an expression depends inversely on a cubed distance variable, fractional variation scales up by three \times the independent tracking variance. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Physics: Units and Measurements Class 12 Physics: Magnetism and Matter