Electromagnetic therory 2023 pyq aktu dbrau with solution

 

✅ SECTION A (All compulsory)

1. Biot-Savart Law
Current element $I\,d\vec{l}$ se magnetic field:

$$d\vec{B} = \frac{\mu_0}{4\pi} \frac{I\, d\vec{l} \times \hat{r}}{r^2}$$

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2. Curl of electric field:
👉 Answer: (b) Rotational

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3. Electric field (E = vB)

$$E = vB = 5 \times 3.6 = 18$$

👉 Answer: (b) 18

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4. Del operator (∇):
👉 Used for Gradient, Divergence, Curl
👉 Answer: (d) Velocity differential operator (best match)

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5. Scalar factor in Cartesian system:
👉 True

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6. Stokes theorem uses:
👉 Answer: (c) Curl

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7. Electric field intensity definition:

$$E = \frac{F}{q}$$

👉 Answer: (c)

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8. Statement:
👉 False (Electric field is maximum on surface, magnetic depends on current)

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9. Vector conversion
Given A = 3i − 2j − 4k at (2,3,3)
Correct option:
👉 (a) −3.6j − 4k

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10. Time varying currents produce:
👉 Answer: (c) Electromagnetic

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✅ SECTION B

1. Faraday’s Law

• Changing magnetic flux induces EMF

$$e = -\frac{d\Phi}{dt}$$

• Negative sign → Lenz’s Law

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2. Total charge calculation

Given:

$$\rho_v = 10z^2 x \sin(\pi y)$$ $$Q = \iiint \rho_v \, dV$$ $$Q = \int_{-1}^{2} \int_{0}^{1} \int_{3}^{3.6} 10z^2 x \sin(\pi y)\, dz\, dy\, dx$$

👉 Final result:

$$Q = 10 \cdot \left(\frac{3^2.6 - 3^3}{3}\right)\cdot \left(\frac{2^2 - (-1)^2}{2}\right)\cdot \left(\frac{2}{\pi}\right)$$

(Exam me steps likhna enough hai)

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3(a) Magnetic flux density

• Symbol: B
• Unit: Tesla (T)
• Relation:

$$B = \mu H$$

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3(b) Coulomb’s Law

$$F = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r^2}$$

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4(a) Infinite sheet charge

$$E = \frac{\sigma}{2\epsilon_0}$$

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4(b) Triple product

Given:

$$A = 2a_x - a_y,\; B = 2a_x - a_y + 2a_z,\; C = 2a_x - 3a_y + a_z$$

👉 Scalar triple:

$$A \cdot (B \times C)$$

👉 Vector triple:

$$A \times (B \times C)$$

(Determinant method likho → marks milenge)

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5(a) Cylindrical coordinates

• Coordinates: $(r, \phi, z)$
• Unit vectors: $a_r, a_\phi, a_z$

Differentials:

• Length: $dl = dr\,a_r + r d\phi\,a_\phi + dz\,a_z$
• Area & volume bhi likh sakte ho for extra marks

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5(b) Electric field on sphere

$$E = \frac{1}{4\pi\epsilon_0} \frac{Q}{r^2}$$ $$E = \frac{9\times10^9 \times 6\times10^{-6}}{5^2}$$ $$E = 2160 \, \text{N/C}$$

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✅ SECTION C

1(a) Characteristic impedance

$$Z_0 = \sqrt{Z_{oc} \cdot Z_{sc}}$$ $$Z_0 = \sqrt{750∠-30^\circ \times 600∠-20^\circ}$$ $$Z_0 = \sqrt{450000 ∠ -50^\circ}$$ $$Z_0 = 670 ∠ -25^\circ \, \Omega$$

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1(b) Gauss Law

$$\oint \vec{E}\cdot d\vec{A} = \frac{Q}{\epsilon_0}$$

Differential form:

$$\nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0}$$

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2(a) Short Notes

(i) VSWR

$$VSWR = \frac{V_{max}}{V_{min}}$$

(ii) Skin Effect

• High frequency → current surface pe flow karta hai

(iii) TE Waves

• Electric field transverse, $E_z = 0$

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2(b) Force on charges

Use:

$$F = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r^2}$$

👉 Triangle symmetric hai → forces vector add karo
👉 Final answer: Equal magnitude, 120° direction

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3. Maxwell Equations

Integral form:

1. $\oint E \cdot dA = \frac{Q}{\epsilon_0}$
2. $\oint B \cdot dA = 0$
3. $\oint E \cdot dl = -\frac{d\Phi_B}{dt}$
4. $\oint H \cdot dl = I + \frac{dD}{dt}$

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Differential form:

1. $\nabla \cdot E = \frac{\rho}{\epsilon_0}$
2. $\nabla \cdot B = 0$
3. $\nabla \times E = -\frac{\partial B}{\partial t}$
4. $\nabla \times H = J + \frac{\partial D}{\partial t}$