Ampere Law & Maxwell - Free MCQ Practice Test with solutions, GATE EE Electromagnetic

Answer: d
Explanation: Ampere law states that the line integral of H about any closed path is exactly equal to the direct current enclosed by that path. ∫ H.dl = I The point form will be Curl (H) = J.

Answer: c
Explanation: The proof of the Ampere’s circuital law is obtained from Stoke’s theorem for H and J only.

Answer: a
Explanation: At the conductor-free space boundary, electric field will be maximum and magnetic field will be minimum. This implies electric field is zero inside the conductor and increases as the radius increases and the magnetic field is zero outside the conductor and decreases as it approaches the conductor.

Answer: b
Explanation: The magnetic field intensity is given by H = I/2πr, where I = 3A and r = 0.12. The magnetic flux density in air B = μ H, where μ = 4π x 10^-7.Thus B = 4π x 10^-7 x 3/2π x 0.12 = 5x 10^-6 units.

Answer: c
Explanation: The magnetic field intensity is given by H = NI/2πrm, where N = 50, I = 2A and rm = 1/2π. Thus H = 50 x 2/2π x 0.159 = 100 units.

Answer: c
Explanation: The magnetic intensity when the normal component is below the sheet is Hy = -0.5 K, where K = 12.Thus we get H = -0.5 x 12 = -6 units.

Answer: b
Explanation: By Ampere law, Curl (H) = J. The curl of H will be i(-cos x) – j(0) + k(-z sin x) = -cos x i – zsin x k. In the xy plane, z = 0. Thus Curl(H) = J = -cos x i.

Answer: b
Explanation: By Ampere law, Curl(H) = J. The rotational path of H is zero, implies the curl of H is zero. This shows the current density J is also zero. The current is the product of the current density and area, which is also zero.

Answer: a
Explanation: We know that ∫ H.dl = I. By Stoke’s law, we can write Curl(H) = J. In integral form, H = ∫ J.ds, where J = 0.5 and ds is defined by 20 units. Thus H = 0.5 x 20 = 10 units.

Answer: d
Explanation: The divergence of the magnetic flux density is always zero. This is because of the non existence of magnetic monopoles in a magnetic field.

Answer: b
Explanation: The charge density is the divergence of the electric flux density by Maxwell’s equation. Thus ρ = Div (D) and Div (D) = 2 + 3 + 4 = 9. We get ρ = 9 units.

Answer: c
Explanation: From the Faraday’s law and Lenz law, using Stoke’s theorem, we get Curl(E) = -dB/dt. This is the Maxwell’s first law of electromagnetics.

Answer: c
Explanation: From the current density definition and Ohm’s law, the Ampere circuital law Curl(H) = J can be derived. This is Maxwell’s second law of electromagnetics.

Answer: a
Explanation: The stationary loop in a varying magnetic field results in an induced emf due to the change in the flux linkage of the loop. This emf is called as induced or transformer EMF.

Answer: a
Explanation: Maxwell equations can be represented in differential/point form and integral form alternatively. Sometimes, it can be represented by time varying fields called harmonic form.

Answer: b
Explanation: The charge in the capacitor is due to displacement current. It is the current in the presence of the dielectric placed between two parallel metal plates.

Answer: a
Explanation: Generally, the Curl(H) is the sum of two currents- conduction and displacement. In case of metals, it constitutes conduction J and in case of dielectrics, it constitutes the displacement current dD/dt.

Answer: a
Explanation: The total flux in a material is the product of the flux density and the area. It is given by flux = 12 x 0.8= 9.6 units.

Answer: c
Explanation: The electric flux density from Maxwell’s equation is given by D = ∫ ρ dv. On substituting ρ = 16 and ∫dv = 1, we get D = 16 units.

CONCEPT:

• Ampere’s Law: Line integral of the magnetic field B around any closed curve is equal to μ₀ times the net current I threading through the area enclosed by the curve i.e.
  
• The intensity of the magnetic field due to wire of infinite length is
  

Where μ₀ = permittivity of free space, I = current in a wire, d = distance

CALCULATION:
Given – I = 15 A, d₁ = 10 cm =10 × 10^-2 m and d₂ = 5 cm = 5 × 10^-2 m
The intensity of the magnetic at P field due to 1^st wire is

The direction of B₁ will be toward point P
The intensity of the magnetic at P field due to 2^nd wire of infinite length is

The direction of B₂ will be toward point P.
∴ The net magnetic field at point P will be the addition of the two, i.e.
B_net = B₂ + B₁ = 6 × 10^-5 + 3 ×10^-5 = 9 × 10^-5 T