a) What mathematical requirement is needed for a vector field to be conservative? b) What is the...

Question:

a) What mathematical requirement is needed for a vector field to be conservative?

b) What is the definition of the circulation of a vector field E for a given loop C?

c) When measuring voltage why must you always use 2 leads on your voltmeter?

d) How much energy does a 12V battery use to push 2 Coulombs of charge through a pair of headlights?

Calculus - Circulation and gradient

Gradient is a function that tells you how fast a function is changing its value. Gradient is a multivariable operator and its out put is a vector. Derivative also gives the change in the value of a function, but the output is a scalar quantity. The gradient takes its input as scalar fields. Gradient theorem or fundamental theorem of calculus helps us to define scalar potential function. Circulation as the name indicates is all about circulation of a vector field. The circulation ensures a vortex or sink. If circulation is non zero a vector field there exists a nonzero torque.

Part a)

For a vector field to be conservative, mathematically there are three conditions.

1) A vector field E will be conservative if the vector field can be expressed as gradient of a scalar potential function V

Then we must have a scalar potential V such {eq}E = \nabla V {/eq}

2) The integral of the vector field over a closed path must be equal to zero {eq}\oint E \cdot dr = 0 {/eq}

The first two conditions tells that conservative field must be path independent.

3) The curl of the vector field must be zero. {eq}\nabla \times E = 0 {/eq}

This says that the conservative field must have zero curl or circulation or the conservative field must not be rotational.

Path b)

Circulation of a vector field gives information about the vector field that it circulates around some point or aligns along some given curve. Circulation of the vector field can be defined as the line integral of the vector field along a curve. Mathematically this can be expressed as {eq}\int_{C} F\cdot ds {/eq} - means the circulation of the vector field F around the curve C. Circulation is fluid mechanics is defined as the rotation of a finite area in fluid

Part c

We can tell that a electric potential at a point, but when we have to measure it we have to measure it with respect to some reference point. So we are expressing potential as potential difference with respect to some reference point. The potential we measure is not absolute potential but rather relative.

Part d)

The work done is moving a charge of 2 coulomb over the light bulb by a 12 volt battery is equal to the product of charge and potential difference.

So the work done {eq}W = q V \\ W = 2 \times 12 \\ W = 24 \ \ J {/eq} 