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To find the force between two charges, we can use Coulomb’s Law, which is expressed as:

[F = k cdot frac{|q_1 cdot q_2|}{r^2}]

where:

– (F) is the force between the charges,

– (k) is Coulomb’s constant ((8.987 times 10^9 N m^2/C^2)),

– (q_1) and (q_2) are the magnitudes of the two charges (in this case, 2nC and -1nC, which are (2 times 10^{-9}C) and (-1 times 10^{-9}C) respectively),

– and (r) is the distance between the charges (4cm, which needs to be converted to meters, so (r = 0.04 m)).

Substituting the given values:

[F = 8.987 times 10^9 N m^2/C^2 cdot frac{(2 times 10^{-9} C) cdot (1 times 10^{-9} C)}{(0.04 m)^2}]

[F = 8.987 times 10^9 cdot frac{2 times 10^{-18}}{0.0016}]

[F = 8.987 times 10^9 cdot 1.

The direction of the force between two charges, such as 1C and -4C existing in air, can be determined using Coulomb’s law, which states that like charges repel each other and unlike charges attract each other. Since one charge is positive (1C) and the other is negative (-4C), the force between them would be attractive. Thus, the direction of the force on each charge would be towards the other charge. Specifically, the 1C charge would experience a force in the direction of the -4C charge, and similarly, the -4C charge would experience a force in the direction of the 1C charge.

To find the divergence of the vector field (mathbf{F} = 4x mathbf{i} + 7y mathbf{j} + z mathbf{k}) and apply the divergence theorem to the specific geometry given (a cone with radius (frac{1}{2pi}) m and height (4pi^2) m), we first calculate the divergence of (mathbf{F}).

The divergence of a vector field (mathbf{F} = Pmathbf{i} + Qmathbf{j} + Rmathbf{k}) is given by:

[ nabla cdot mathbf{F} = frac{partial P}{partial x} + frac{partial Q}{partial y} + frac{partial R}{partial z} ]

For (mathbf{F} = 4x mathbf{i} + 7y mathbf{j} + z mathbf{k}), we have:

[ nabla cdot mathbf{F} = frac{partial (4x)}{partial x} + frac{partial (7y)}{partial y} + frac{partial (z)}{partial z} = 4 + 7 + 1 = 12 ]

Now, the divergence theorem states that for a vector field (mathbf{F}

To apply the divergence theorem to the function (f(x, y, z) = x^2 + y^2 + z^2) at a distance of one unit from the origin, we first need to understand what the divergence theorem is and how it’s used in this context.

The divergence theorem, also known as Gauss’s theorem, relates the flow (flux) of a vector field through a closed surface to the divergence of the vector field inside the volume enclosed by the surface. Mathematically, it can be expressed as:

[

intint_{Sigma} vec{F} cdot dvec{A} = intintint_{V} nabla cdot vec{F} dV

]

where (Sigma) is the closed surface, (V) is the volume enclosed by (Sigma), (vec{F}) is a vector field, (dvec{A}) is a vector element of area on (Sigma), and (dV) is an element of volume inside (V).

For the function (f(x, y, z) = x^2 + y^2 + z^2), we first convert this scalar function into a vector field. A common way to do this, given the context is not explicitly directing towards a specific vector field construction, is to consider a gradient field derived from (f), since the divergence theorem