Anna University - Previous year Question Bank - Part B, EMF

1. UNIT WISE ANNA UNIVERSITY QUESTIONS
PART – B QUESTIONS
Faculty Name: Mr. K.SIVAKUMAR Branch & Sem: ECE & IV
Subject Name: Electromagnetic Fields Code:EC8451
No. Unit-I:: STATIC ELECTRIC FIELD
1
2
i. Given the two points A(x=2,y=3,z=-1) and B(r=4,𝜃 = 250,∅ = 120𝑜find the spherical co-
ordinates of A and Cartesian Co- ordinates of B
ii. Find ∇𝑥𝐻
⃗
⃗ , if 𝐻
̅ = (2𝜌𝑐𝑜𝑠∅𝑎𝜌
̅̅̅ − 4𝜌𝑠𝑖𝑛∅𝑎∅
̅̅̅ + 3𝑎𝑧
̅̅̅)
i. A circular disc of radius ‘a’ m is charged uniformly with a charge of σ c/𝑚2. Find the
electric field intensity at a point ‘h’ meter from disc along its axis.
ii. If 𝑉 = [2𝑥2𝑦 + 20𝑧 −
4
𝑥2+𝑦2
] volts, find E and D at 𝑃(6, −2.5,3).
April
2010
1
2
(i) Find the total electric field at the origin due to 10−8 C charge located at 𝑃 (0, 4,4) m and a -
0.5×10−8 C charge at P (4, 0, 2) m.
(ii) Derive an expression for the electric field intensity at any point due to a uniformly charged
sheet with density ρ𝑠 c/𝑚2.
(i) Point charges Q and –Q are located at (0,0,
𝑑
2
) and (0,0,-
𝑑
2
). show that the potential at a
point (r,θ,φ) is inversely proportional to r2
noting that r>>d.
(ii) Given a field E= 𝐸 =
−6𝑦
𝑥2
𝑎𝑥 +
6
𝑥
𝑎𝑦 + 5𝑎𝑧 V/m, find the potential difference VAB between A(-
7,2,1) and B(4,1,2).
April
2011
1
2
State and explain the boundary conditions of electric field at dielectric and conductor.
Deduce an expression for the capacitance of a parallel plate capacitor with two dielectrics of
relative permittivity 𝜀1 and 𝜀2 respectively interposed between the plates. April
2012
1
2
Derive an expression for the electric field due to the straight and infinitely uniformed charged
wire of length ‘L’ meters and with a charge density of +𝜌 𝐶/𝑚 at a point P which lies along
perpendicular bisector of wire.
i. A uniform line charge 𝜌𝐿=25𝑁𝑐/𝑚 lies on the 𝑥 = 3𝑚 and 𝑦 = 4𝑚 in the free space. Find the
electric field intensity at a point (2,3 𝑎𝑛𝑑 15) m.
ii. Given the potential 𝑉 = 10𝑠𝑖𝑛𝜃𝑐𝑜𝑠𝜑/𝑟2 find the electric flux density D at (2,𝜋/2,0).
April
2013

2. 1
2
(i) Find the electric field intensity at point P located at (0,0,h)m due to surface charge density σ
c/𝑚2uniformly distributed over circular disc r ≤ a, z=0m.
(ii) Determine the divergence and curl of the given field F=30𝑎𝑥+2xy𝑎𝑦+5x𝑧2𝑎𝑧 at (1,-1, 0.2) and
hence state the nature of field.
(i) Derive the expression for potential due to an electric dipole at any point P. Also find the
electric field intensity at the same point.
(ii) Two point charges, 1.5nC at (0, 0, 0.1) and -1.5nC at (0, 0, -0.1) are in free space. Treat the
two charges as a dipole at the origin and find the potential at point P (0.3, 0, 0.4)
Dec2010
1
2
(i) Assume a straight line charge extending along the z-axis in a cylindrical Co-ordinate system
from −∞ 𝑡𝑜 ∞. Determine the electric field intensity 𝐸
̅ at every point resulting from a uniform line
charge density 𝜌𝐿 𝐶/𝑚.
(ii) Consider an infinite uniform line charge of 5𝑛𝐶/𝑚 parallel to z-axis at𝑥 = 4,𝑦 = 6. Find the
electric field intensity at the point 𝑃(0,0, 5) in free space.
(i) The flux density within the cylindrical volume bounded by r=2m, z=0 and z=5m is given by
𝐷
̅ = 30𝑒−𝑟𝑎𝑟
̅̅
̅̅ -2z𝑎𝑧
̅̅̅ C/𝑚2 . What is the total outward flux crossing the surface of the cylinder.
(ii) State and prove Gauss law for electric field. Also give the differential form of Gauss law.
Dec2011
1
2
(i)find the intensity at a point P located at (0,0,h)m due to charge of surface charge density
(ii) Determine the divergence and curl of the given field F=30ax + 2xyay +5xz2
az at a(1,1,-0.2) and
hence state the nature of the field.
(i) Point charges Q and –Q are located at (0,0,
𝑑
2
) and (0,0,-
𝑑
2
) show that the potential at a point
(r,θ,φ) is inversely proportional to r2
noting that r>>d.
(ii) Given a field E= 𝐸 =
−6𝑦
𝑥2
𝑎𝑥 +
6
𝑥
𝑎𝑦 + 5𝑎𝑧 V/m, find the potential difference VAB between A(-
7,2,1) and B(4,1,2).
Dec2012
1
2
Apply Gauss law to find charge enclosed in hollow sphere whose surface is uniformly charged.
Derive the equation for potential due to a system of point charge.
State and prove stoke’s theorem and divergence theorem.
Dec2013
State and explain the fundamental theorems of divergence and curl
May
2014
No. Unit-II:: CONDUCTORS AND DIELECTRICS

3. 1 (i) Derive Poisson’s and Laplace’s equation.
(ii) A parallel plate capacitor has an area of 0.8 m2, separation of 0.1 mm with a dielectric for
which 1000=𝜀𝑟 and a field of 610 V/m. Calculate C and V.
April
2010
1 (i) Determine whether or not the following potential fields satisfy the Laplace’s equation.
I. V= x2
– y2
+ z2
II. V= r cosφ + z
III. V= r cos θ + 𝜑
(ii) Solve the Laplace’s equation for the potential field in the homogenous region between the
two concentric conducting spheres with radius ‘a’ and ‘b’ where b>a , V=0 and r=b and V=Vo at
r=a. Find the capacitance between the two concentric spheres.
April
2011
1
2
State and derive Poisson’s equation and Laplace equation.
Obtain the expression for energy stored in magnetic field and also derive an expression for
magnetic energy density.
April
2012
1
2
Derive the boundary condition of normal and tangential components of electric field at an
interface of two media with different dielectrics.
April
2013
1
2
(i) Write the Poisson’s and Laplace’s equation.
(ii) Discuss the magnetic boundary conditions.
(i) To concentric metal spherical shells of radii a and b are separated by weakly conducting
material of conductivity σ. If they are maintained at a potential difference V. What current flows
from one to the other? What is the resistance between the shells? Find the resistance if b>>a.
April
2014
1 (i) Write down the Poisson’s and Laplace’s equation. State their significance in electrostatic
problems.
(ii) Two parallel conducting plates are separated by distance ‘d’ apart and filled with dielectric
medium having εr as relative permittivity. Using Laplace equation, derive an expression for
capacitance per unit length of parallel plate capacitor, if it is connected to a DC source of V volts.
Dec2010
1
2
(i) A metallic sphere of radius 10 cm has a surface charge density of 10 nC/𝑚2 . Calculate the
energy stored in the system.
(ii) State and explain the electric boundary conditions between two dielectrics with
permittivity𝜀1 𝑎𝑛𝑑 𝜀2.
(i) Derive the expression for continuity equation of current in differential form. 11
Dec2011

4. 1 (i) Determine whether or not the following potential fields satisfy the Laplace’s equation.
V= x2
– y2
+ z2
V= r cosφ + z
V= r cos θ + 𝜑
(ii) Solve the Laplace’s equation for the potential field in the homogenous region between the two
concentric conducting spheres with radius ‘a’ and ‘b’ where b>a V=0 and r=b and V=Vo at r=a.
Find the capacitance between the two concentric spheres.
Dec2012
1
2
Derive the boundary relations for
(i) E-field
(ii) H-field.
A composite conductor of cylindrical cross section used in overhead line is made of steel inner
wire of radius “a” and an annular outer conductor of radius “b”, the two having electrical contact.
Evaluate the H-field within the conductors and the internal self-inductance per unit length of the
conductor.
Dec2013
No. Unit-III:: STATIC MAGNETIC FIELDS
1 (i) State and explain Ampere’s circuital law. (8)
(ii) Find an expression for H at any point due to a long, straight conductor carrying I amperes.
April
2010
1 (i) Using Biot-Savart’s law, derive the magnetic field intensity on the axis of a circular loop
carrying a steady current I. (ii) Using
Ampere’s circuital law, derive the magnetic field intensity due to a co-axial cable carrying a
steady current I. .
April
2011
1 State and explain Ampere’s Circuital law. Show that the magnetic field at the end of a long
solenoid is one half that at the centre.
April
2012
1 (i) Derive an expression for force between two current carrying conductors.
(ii) An iron ring with a cross sectional area of 3cm square and mean circumference of 15cm is
wound with 250 turns wire carrying a current of 0.3A. The relative permeability of the ring is 1500.
Calculate the flux established in the ring.
April
2013
1 (i) Derive the expression for magnetic field intensity due to a linear conductor of infinite length
carrying current I at a distant point P. Assume R to be the distance between the conductor and
point P. Use Biot-Savart’s law. (8)
(ii) Derive the expression for magnetic field intensity on the axis of circular loop of radius ‘a’
carrying current I.
Dec2010

5. 1
2
(i) Find the magnetic field intensity due to a finite wire of carrying a current I and hence deduce
an expression for magnetic field intensity at the centre of a square loop.
(ii) A circular loop located on 𝑥2 + 𝑦2= 4, z=0 carries a direct current of 7A along 𝑎𝜑
̂. Find the
magnetic field intensity at (0, 0, -5).
(i) Using Ampere’s circuital law determine the magnetic field intensity due to a infinite long wire
carrying a current I.
Dec2011
1 (i) Derive an expression for magnetic field intensity due to a linear conductor of infinite length
carrying current I at a distance point P. Assume R to be the distance between conductor and
point P. Use Biot-savart’s l
(ii) Derive an expression for magnetic field intensity on the axis of a circular loop of radius ‘a’
carrying current I.
Dec2012
1
2
Derive the expression for Biot-Savart law. Derive a equation for torque on a current carrying loop.
Find H-field on the axis of a ring carrying a constant current. Highlight the similarities between
Biot-Savart law and coulomb’s law
Dec2013
No. Unit-IV:: MAGNETIC FORCES AND MATERIALS
1
2
(i) Find the maximum torque on an 85 turns, rectangular coil with dimension 0.2×0.3 m, carrying
a current of 5 Amps in a field B=6.5T
(ii) Derive an expression for Magnetic vector potential
(i) Derive an expression for the inductance of solenoid.
(ii) Derive the boundary conditions at an interface between two magnetic Medias.
April
2010
1 (i) Derive an expression for a torque on a closed rectangular loop carrying current.
(ii) In cylindrical co-ordinates, 𝐴̅ = 50𝜌2𝑎𝑧
̅̅̅ Wb/m is a vector magnetic potential in a certain region
of free space. Find the magnetic field intensity (H), magnetic flux density (B) and current density
(J).
April
2011
1
2
A solenoid with radius of 2cms is wound with 20 turns per cm and carries 10 mA. Find H at the
centre of the solenoid if its length is 10 cm. If all the turns of the solenoid were compressed to
make a ring of radius 2cms, what would be H at the centre of the ring?
Obtain the expression for energy stored in magnetic field and also derive an expression for
magnetic energy density.
April
2012

6. 1
2
(i) Obtain the expressions for scalar and vector magnetic potentials (8)
(ii) The vector magnetic potential 𝐴 =(3y-3) 𝑎 𝑥 +2xy𝑎 𝑦 Wb/m in a certain region of free space
1. Show that ∇.𝐴 =0.
2. Find the Magnetic flux density 𝐵
⃗ and the Magnetic field intensity 𝐻
⃗
⃗ at
Point P(2, -1, 3)
(i) Derive an expression for the inductance of toroidal coil carrying current.
(ii) A solenoid is 50cm long, 2cm in diameter and it contains 1500 turns. The cylindrical core has
a diameter of 2cm and a relative permeability of 75. The coil is co axial with a second solenoid,
also 50cm long, but 3cm diameter and 1200 turns. Calculate the L for inner and outer solenoid.
Dec2010
1 (i) Derive an expression for inductance of a solenoid with N turns and l metre length carrying a
current a current of I amperes.
(ii) An iron ring of relative permeability 100 is wound uniformly with two coils of 100 and 400 turns
of wire. The cross section of the ring is 4𝑐𝑚2 . The mean circumference is 50 cm. Calculate
The self inductance of each of the two coils 1.The mutual inductance 2.Total inductance when
the coils are connected in series with flux in the same sense.3.Total inductance when the coils
are connected in series with flux in the opposite sense.
Dec2011
1 (i) derive the inductance of the toroidal coil with N turns, carrying current I and the radius of the
toroid R.
(ii) Considering a toroidal coil, derive an expression for energy density.
Dec2012
No. Unit-V:: TIME VARYING FIELDS AND MAXWELL’S EQUATIONS
1
2
Derive the Maxwell’s equations derived from Faradays law in both integral and point form.
Derive modified form of Ampere’s circuital law in Integral and differential Forms.
April
2010
1
2
(i) State and prove Poynting theorem.
(ii) Derive the expression for total power flow in co-axial cable.
(i) From the Maxwell’s equation, derive the electromagnetic wave equation in conducting medium
for E and H fields.
(ii) Calculate the attenuation constant and
April
2011

7. 1
2
3
4
5
State and derive the Maxwell’s equation for free space in integral and point forms for time varying
fields.
State and derive the Maxwell’s equation for free space in integral and point forms for time varying
fields.
State and prove boundary conditions by the application of Maxwell’s equations.
Find the amplitude of the displacement current density inside a capacitor where 𝜀𝑟 = 600 and
D= 3× 10−6sin(6 × 106 𝑡 − 0.3464 𝑦)𝑢𝑧
̅̅̅c/𝑚2.
Obtain standing wave equation when electromagnetic wave incident normally on a perfect
conductor.
April
2012
1
2
With necessary explanations, derive Maxwell’s equation in differential and integral forms.
(i)The conduction current flowing through a wire with conductivity σ=3x107
s/m and the relative
permeability εr=1 is given by Ic=3sinωt(mA). If ω=108
rad/sec. Find the displacement current.
(ii) An electric field in a medium which is source free is given by E=1.5cos (108
t-βz)ᾶx V/m. Find
B,H and D. Assume εr=1,µr=1,σ=0.
April
2013
1
2
(i) Explain Ampere’s circuital law.
(ii) Derive Poynting’s theorem.
(i) Describe the Maxwell’s equation in differential and integral forms.
April
2014
1
2
(i) Explain Ampere’s circuital law.
(ii) Derive Poynting’s theorem.
(i) Describe the Maxwell’s equation in differential and integral forms.
Dec2010
1
2
3
4
5
Give the physical interpretation of Maxwell’s first and second equation.
State and prove Poynting theorem.
In free space, 𝐸
̅ = 50cos(𝜔𝑡 − 𝛽𝑧)𝑎𝑧
̅̅̅ V/m. Find the average power crossing a circular area of
radius 2.5 m in the plane z=0. Assume 𝐸𝑚 = 𝐻𝑚𝑛𝑜 and 𝑛𝑜 = 120𝜋.
From the Maxwell’s equation, derive the electromagnetic wave equation in conducting medium
for E and H fields.
The electric field associated with a plane wave travelling in a perfect dielectric medium is given
by 𝐸𝑥(𝑧,𝑡) = 10cos[2𝜋 × 107 − 0.1𝜋 𝑥] V/m. Find the velocity of propagation and intrinsic
impedance. Assume 𝜇 =𝜇𝑜 .
Dec2011

8. 1
2
(i) State and prove Poynting theorem.
(ii) Derive the expression for total power flow in co-axial cable.
(i) From the Maxwell’s equation, derive the electromagnetic wave equation in conducting medium
for E and H fields.
(ii) Calculate the attenuation constant and phase constant for the uniform plane wave with the
frequency of 100GHz in a conducting medium for which 𝜇𝑟 = 1 and 𝜎 = 58 × 106 𝑆/𝑚 . (6)
Dec2012
1
2
State and prove pointing theorem. Write the expression for instantaneous average and complex
Poynting vector.
Write the inconsistency of ampere’s law. Is it possible to construct a generator of EMF which is
constant and does not vary with time by using EM induction principle? Explain.
Derive the wave equations from Maxwell’s equations. Give the illustration for plane waves in
good conductors.
Dec2013