Test: Divergence

Answer: a
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).

Answer: a
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.

Answer: d
Explanation: The vector given is a position vector. The divergence of any position vector is always 3.

Answer: b
Explanation: Div (yi + zj + xk) = Dx(y) + Dy(z) + Dz(x), which is zero. Here D refers to partial differentiation.

Answer: d
Explanation: Div (D) = Dx(e^-xsin y) + Dy(-e^-xcos y ) = -e^-xsin y + e^-xsin y = 0.

Answer: a
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).

Answer: b
Explanation: Div(F) = Dx(30) + Dy(2xy) + Dz(5xz²) = 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 b, as by theory when the divergence is zero, the vector is solenoidal. Option b is the only one which is satisfying this condition.

Answer: a
Explanation: Div(E) = Dx(yz) + Dy(xz) + Dz(xy) = 0. The divergence is zero, thus vector is divergentless or solenoidal.

Answer: b
Explanation: Div(P) = Dx(x²yz) + Dy(0) + Dz(xz) = 2xyz + x, which is option b. For different values of x,y,z the divergence of the field varies.

Since the vector field does not diverge (moves in a straight path), the divergence is zero. Also, the path does not possess any curls, so the field is irrotational.