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Q1)
Given E̅ = −3x𝑦 3 𝑧 2 𝑖 + 6𝑥 3 𝑦 2 𝑧𝑗 − 8𝑥 2 𝑦 2 𝑧 2 𝑘
The surface is given by
𝑥: 0 → 1
𝑦: 0 → 2
𝑧: 0 → 3
According to divergence theorem,
∬ 𝐸. 𝑑𝑆 = ∭ 𝑑𝑖𝑣𝐸 𝑑𝑉
𝑑𝑖𝑣𝐸 = ∇. 𝐸 = −3𝑦3 𝑧2 + 12𝑥3 yz − 16𝑥2 𝑦2 𝑧
Where
∴
∬ 𝐸. 𝑑𝑆 = ∭ 𝑑𝑖𝑣𝐸 𝑑𝑉 = ∭ −3𝑦3 𝑧2 + 12𝑥3 yz − 16𝑥2 𝑦2 𝑧 𝑑𝑥 𝑑𝑦 𝑧
Upon integrating by each variable (x, y & z) and substituting the surface boundary values,
We get
16
∬ 𝐸. 𝑑𝑆 = ∬ −3𝑦3 𝑧2 + 3𝑦𝑧 − 3 × 𝑦2 𝑧 𝑑𝑦 𝑑𝑧
= ∫ −12𝑧2 + 6𝑧 −
16×8
9
× 𝑧 𝑑𝑧
∬ 𝐸. 𝑑𝑆 = −4 × 33 + 27 − 64 = −𝟏𝟒𝟓
Q2)
a)
SIGNIFICANCE OF CONSERVATIVE VECTOR
In electrostatics, conservative vectors play a significant role in understanding and analyzing
the behavior of electric fields. A conservative vector is a vector field where the work done in
moving along a closed path is independent of the path taken but depends only on the initial
and final points. In the context of electrostatics, this means that the electric field produced by
conservative vectors possesses certain fundamental properties.
One crucial property of conservative vector fields in electrostatics is that they can be
expressed as the gradient of a scalar function, known as the electric potential. The electric
potential is a scalar quantity which represents the amount of work done per unit charge in
moving a test charge from a reference point another location in the field. By taking the
gradient of the electric potential, one can obtain the electric field. Conservative vectors also
have another significant implication in electrostatics: the conservation of energy. Due to the
conservative nature of the electric field, the work done by the field in moving a charged
particle along any closed loop is zero.Moreover, conservative vector fields have
mathematical properties that simplify the calculation of electric fields. For example, Gauss's
law in electrostatics relies on the conservative nature of the electric field to relate the electric
flux through a closed surface to the net charge enclosed by that surface.
In conclusion, we can say that the significance of conservative vectors in electrostatics lies in
their connection to the electric potential, the conservation of energy, and their role in
simplifying the mathematical analysis of electric fields
Given vector field is given by,
̅ = (6xycos z)𝑖 + ( 3 𝑥 2 cosz) 𝑗 − (3𝑥 2 y sin z )𝑘
A
̅ = (M) 𝑖 + (N) 𝑗 + (P) 𝑘
A
Similar to
̅ is said to be conservative if and only if
A
𝜕𝑀
𝜕𝑦
𝜕𝑁
𝜕𝑁
= 𝜕𝑥
𝜕𝑧
𝜕𝑃
𝜕𝑀
= 𝜕𝑦
𝜕𝑧
−3𝑥 2 sin z = 6𝑥𝑐𝑜𝑠𝑧
6𝑥𝑐𝑜𝑠𝑧 = 6𝑥𝑐𝑜𝑠𝑧
−6xysin z = −6xysin z
So the given vector filed in conservative.
b)
Given
Thus
,
Frequency,𝜗 = 200 MHz
𝜔 = 2×𝜋×𝜗
𝜇𝑟 = 1
Loss tangent, 𝜔𝜀 = 3.4 × 10−3
𝜇 = 𝜇0
𝜖𝑟 = 7
𝜎
𝜖 = 7 × 𝜖0
𝜇𝜖
𝜎
2
Phase shift constant, 𝛽 = 𝜔√ 2 [√1 + (𝜔𝜀) + 1]
Upon substituting values, 𝜷 = 𝟏. 𝟕𝟔𝟓 𝒓𝒂𝒅/𝒎
Intrinsic impedance is given by 𝜂 = |𝜂|∠𝜃𝑛
𝜕𝑃
= 𝜕𝑥
𝜇
Where |𝜂| =
√𝜖
1
𝜎 2 4
[1+( ) ]
𝜔𝜀
&
1
𝜎
𝜃𝑛 = 2 × 𝑡𝑎𝑛−1 (𝜔𝜀)
Upon substituting the values, |𝜂|= 142.424 Ω
𝟏𝟒𝟐. 𝟒𝟐𝟒∠𝟎. 𝟎𝟗𝟕𝟒
Q3)
&
𝜃𝑛 =0.0974
∴𝜼=
a)
Given
Magnetic field intensity, 𝐻 = 20 𝐴/𝑚
Intensity of magnetisation, 𝐼 = 318𝐴/𝑚
𝐼
318
Then Magnetic susceptibility, 𝑋𝑚 = 𝐻 = 20 = 15.9
We have
relative permeability,
𝜇𝑟 = 1 + 𝑋𝑚
𝝁𝒓 = 𝟏𝟔. 𝟗
Permeability, 𝜇 = 𝜇0 × 𝜇𝑟 = 𝟐𝟏𝟐. 𝟑𝟕𝟐 × 𝟏𝟎𝟑 𝑇𝑚/𝑎𝑚𝑝
Since this material have𝝁𝒓 ≫ 𝟏, The material is ferromagnetic.
b)
Given
𝐵̅ = −2𝑖 + 𝑗 − 𝑘
̅ in x direction, & 𝑀
̅ = 𝑁𝐼𝐴 𝑖 Where N = Number of turns, I= Curre...