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The correct choice is (a) True

To explain I WOULD say: GAUSS law states that the electric flux PASSING through any closed SURFACE is equal to the total charge enclosed by the surface. Thus the charge is defined as a surface INTEGRAL.

Right option is (B) 0

The explanation is: WORK done in a lamellar FIELD is zero. ∫ E.dl = 0,thus ∑V = 0. The potential will be zero.

The correct CHOICE is (d) Conservative

To elaborate: Work done in moving a charge in a closed PATH is zero. It is expressed as, ∫ E.dl = 0. The FIELD having this property is called conservative or lamellar field.

Correct CHOICE is (a) ln(b/a). State True/False.

The best I can explain: The ELECTRIC FIELD intensity is given by, E = λ/(2πεr)

Vab = -∫ E.dr = -∫ λ/(2πεr). On INTEGRATING from b to a, we get λ/(2πε) ln(b/a).

Right OPTION is (c) -8.214

The BEST I can explain: V = -∫ E.dl = -∫ (-6y/x2 )dx + ( 6/x)dy + 5 dz, from b to a.

On INTEGRATING, we GET -8.214 volts.

The correct answer is (c) 106

Explanation: V = -∫ E.dl = -∫ (40xy dx + 20x^2 DY + 2 dz) , from Q to p.

On INTEGRATING, we get 106 VOLTS.

The CORRECT choice is (b) 57.905

To explain I WOULD SAY: E = -GRAD (V) = -4xy i – 2×2 j + 5k

At (-4,3,6), E = 48 i – 32 j + 5 k, |E| = √3353 = 57.905 units.

The CORRECT CHOICE is (d) WAVEGUIDES

Explanation: In the Directional coupler, Magic Tee, Isolator and Terminator the EM waves travel both in linear and angular motion, which involves curl too. But in waveguides, as the name suggests, only guided PROPAGATION occurs (no bending or curl of waves).

The correct option is (b) 0.72

Easy explanation: W = -Q E.dl

W = -2 X 10^-3 X (6y^2z i + 12xyz j + 6xy^2 K) . (-3 i + 5 j -2 k)

At p(0,2,5), W = -2(-18.22.5) X 10^-3 = 0.72 J.

The correct option is (d) i + yj + (4Y – z)K

Explanation: CURL A = i(Dy(y) – DZ(0)) – j (Dx(0) – Dz(yz)) + k(Dx(4xy) – Dy(yz)) =

i + y j + (4y – z)k.

Correct ANSWER is (b) i – ex J – cos AX k

The best EXPLANATION: Curl A = i(Dy(y + ex)) – j (Dx(y + ex) – Dz(y cos ax)) + k(-Dy(y cos ax))

= 1.i – j(ex) – k cos ax = i – ex j – cos ax k.

Correct option is (a) Yes

To explain: Curl E = i(Dy(XY) – DZ(XZ)) – j (DX(xy) – Dz(yz)) + k(Dx(xz) – Dy(yz)) =

i(x – x) – j(y – y) + k(z – z) = 0

Since the curl is zero, the vector is irrotational or curl-free.

Right option is (b) Vector

The EXPLANATION: CURL is always DEFINED for VECTORS only. The curl of a vector is a vector only. The curl of the resultant vector is also a vector only.

Correct CHOICE is (C) Stoke’s theorem

The explanation: The Stoke’s theorem is given by ∫ A.dl = ∫Curl(A).ds, which uses the curl operation. There can be confusion with MAXWELL equation also, but it uses curl in electromagnetics specifically, WHEREAS the Stoke’s theorem uses it in a generalised manner. Thus the BEST option is Stoke’s theorem.

Correct answer is (B) Grad(DIV V) – (Del)^2V

Easiest explanation: Curl (Curl V) = Grad (Div V) – (Del)^2V is a standard result of the curl operation.

Right OPTION is (b) 0, solenoidal

The best EXPLANATION: Div(F) = Dx(30) + DY(2xy) + Dz(5xz^2) = 0 + 2x + 10xz = 2x + 10xz

DIVERGENCE at (1,1,-0.2) will give zero. As the divergence is zero, field is solenoidal.

Alternate/Shortcut: Without calculation, we can easily choose option “0, solenoidal”, as by theory when the divergence is zero, the vector is solenoidal. “0, solenoidal” is the only one which is satisfying this CONDITION.

Right choice is (a) True

To elaborate: Curl is DEFINED as the circulation of a vector per unit area. It is the cross product of the DEL operator and any vector field. Circulation implies the angular at every point of the vector field. It is obtained by multiplying the component of the vector PARALLEL to the specified closed path at each point ALONG it, by the differential path LENGTH and summing the results.

Correct option is (a) Yes, solenoidal

To explain I would say: Div(E) = Dx(YZ) + Dy(xz) + DZ(XY) = 0. The divergence is zero, thus vector is divergentless or solenoidal.

The correct option is (b) 2xyz + x

The explanation is: Div(P) = DX(x^2yz) + Dy(0) + DZ(xz) = 2xyz + x, which is 2xyz + x. For DIFFERENT values of x, y, z the divergence of the FIELD varies.

The correct OPTION is (a) (1 – x)(1 + e^-x)

Easy explanation: Div(F) = Dx(xe^-x) + DY(y)+Dz(-XZ) = -xe^-x + e^-x + 1 – x =

e^-x(1 – x) + (1 – x) = (1 – x)(1 + e^-x).

Correct choice is (b) 0

Easy explanation: Div (yi + zj + xk) = DX(y) + Dy(Z) + DZ(X), which is zero. Here D refers to partial differentiation.

The CORRECT choice is (a) True

The explanation: CONSIDER the illustration of Pascal’s law, wherein a ball is PRICKED with holes all over its BODY. After water is filled in it and pressure is applied on it, the water flows out the holes uniformly. This is analogous to the flux flowing outside a CLOSED surface as the volume reduces.

Correct answer is (d) 3

The best I can EXPLAIN: The vector GIVEN is a position vector. The DIVERGENCE of any position vector is always 3.

Right option is (a) True

Easy explanation: Divergence can be computed only for a vector. Since it is the MEASURE of OUTWARD flow of FLUX from a small closed surface as the VOLUME shrinks to zero, the result will be directionless (scalar).

Correct CHOICE is (a) True

Best explanation: GRADIENT of a FUNCTION is zero IMPLIES slope is zero. When slope is zero, the function will be PARALLEL to x-axis or y value is constant.

The CORRECT ANSWER is (b) 2i + 2j + 2k

To explain: Grad(x^2+y^2+z^2) = 2XI + 2yj + 2zk. Put x=1, y=1, z=1, the GRADIENT will be 2i + 2j + 2k.

The correct CHOICE is (c) Differential equation

Explanation: Since gradient is the maximum SPACE RATE of CHANGE of flux, it can be REPLACED by differential equations.

The CORRECT ANSWER is (c) Null vector

To explain I would say: Gradient of any function leads to a vector. Similarly curl of that vector gives ANOTHER vector, which is always zero for all constants of the vector. A zero value in vector is always TERMED as null vector(not SIMPLY a zero).

Right answer is (b) 10I + J + 0.135k

The BEST explanation: Grad(t) = 2XY i + x^2 j + e^z k. On substituting p(1,5,-2), we get 10i + j + 0.135k.

The correct CHOICE is (C) Slope

To explain I WOULD say: The gradient is the rate of change of space of FLUX in electromagnetics. This is analogous to the slope in MATHEMATICS.

Correct answer is (b) False

To explain: GRADIENT of any scalar function may be defined as a vector. The vector’s MAGNITUDE and direction are those of the maximum SPACE rate of CHANGE of φ.