Page 65 - Engineering Electromagnetics, 8th Edition

CHAPTER 2 Coulomb’s Law and Electric Field Intensity 47

2.27 Given the electric field E = (4x − 2y)a x − (2x + 4y)a y , find (a) the equation
of the streamline that passes through the point P(2, 3, −4); (b)a unit vector
specifying the direction of E at Q(3, −2, 5).
2.28 An electric dipole (discussed in detail in Section 4.7) consists of two point
charges of equal and opposite magnitude ±Q spaced by distance d.With the
charges along the z axis at positions z =±d/2 (with the positive charge at
the positive z location), the electric field in spherical coordinates is given
3
by E(r,θ) = [Qd/(4π 0 r )][2 cos θa r + sin θa θ ], where r >> d. Using
rectangular coordinates, determine expressions for the vector force on a point
charge of magnitude q (a)at(0, 0, z); (b)at(0, y, 0).

2.29 If E = 20e −5y (cos 5xa x − sin 5xa y ), find (a) |E| at P(π/6, 0.1, 2); (b)a unit
vector in the direction of E at P;(c) the equation of the direction line passing
through P.
2.30 For fields that do not vary with z in cylindrical coordinates, the equations of
the streamlines are obtained by solving the differential equation E p /E φ =
dρ/(ρdφ). Find the equation of the line passing through the point (2, 30 , 0)
◦
for the field E = ρ cos 2φa ρ − ρ sin 2φa φ .