Solved Exercise/Numericals - Gauss's Law - Application.



Q.1 A point charge of 30 nC is located at the origin while plane y = 3 carries charge 10 nC / m2 . Find D at (0, 4, 3)?
Ans:

Electric flux density (D) at point (0, 4, 3) due to a point charge and line charge is given as:

D = DQ + Dρ








= 0.019 (0, 4, 3) + 5ay

= 0.76 ay + 0.057 az + 5ay

= 5.076 ay + 0.057 az



Q.2 A charge distribution in free space has ρv = 2r nC / m3 for 0 < r < 10 m and zero otherwise. Determine E at r = 2m and r = 12m?
Ans:

Gauss’s law states that:






For 0 ≤ r ≤ 10

Dr (4πr2) = ∫ ∫ ∫ 2r (r2 sinθ dr dθ dφ)

Dr (4πr2) = 4π (2r4) / 4














For r ≥ 10

Dr (4πr2) = ∫ ∫ ∫ 2ro (r2 sinθ dr dθ dφ)

Dr (4πr2) = 4π (2ro4) / 4

Dr (4πr2) = 2πro4













Q.3 If D = (2y2 + z)ax + 4xy ay + x az C/m2, find
a) Volume charge density at (-1, 0, 3)
b) The flux through the cube defined by 0 ≤ x ≤ 1,
c) The total charge enclosed by the cube.
 Ans:

a) Volume charge density (ρv) is defined as:

ρv = ∇ . D






   = 4

At point (-1, 0,3) ρv = 4

b) The total charge (Q) enclosed by the cube

Q = ∫v ρv dv

= ∫(x=0)1 ∫(y=0)1 ∫(z=0)1 4x (dxdydz)

= 4 | (x2 / 2) |01 (1) (1)

= 2 C

c) The flux through the cube defined by:

Ψ = Qenc = 2C



Q.4 If the volume charge density (ρv) of a given charge distribution is given by ρ = ρo (a / r) in spherical co-ordinate, determine the electric flux density (D) at any point ?

Ans:

Gauss’s law states that:







Qenc = ∫v ρv dv

= ∫(r=0)r ∫(θ=0)π ∫(φ=0)2π  ρo (a/r) (r2 sinθ dr dθ dφ)

= - ρo a | r2 / 2 |0r (2π) | (cosθ) |0π

= - ρo a (r2 / 2) (2π) ( -2)

= 2 ρo aπr2

Electric flux is given as:











Since,

Ψ = Qenc

Dr (4πr2) = 2 ρo aπr2

D = [ (ρo a) / 2a ] ar


ALSO READ: 

- Gauss's Law - Theory.

- Gauss's Law - Application To a Point charge.

- Gauss's Law - Application To An Infinite Line Charge.

- Gauss's Law - Application To An Infinite Sheet Charge.

- Gauss's Law - Application To a Uniformly Charged Sphere.

- Numericals / Solved Examples - Gauss's Law.

- Scalar Electric Potential / Electrostatic Potential (V).

- Relationship Between Electric Field Intensity (E) and Electrostatic Potential (V).

- Electric Potential Due To a Circular Disk.

- Electric Dipole.

- Numericals / Solved Examples - Electric Potential and Electric Dipole.

- Energy Density In Electrostatic Field / Work Done To Assemble Charges.

- Numericals / Solved Examples - Electrostatic Energy and Energy Density.

- Numericals / Solved Examples - Gauss's law...


- Short Notes/FAQ's

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