A metal wire has a circular cross section with radius 0.800 mm You measure the resistivity of t

A metal wire has a circular cross section with radius 0.800...

A metal wire has a circular cross section with radius $0.800 \mathrm{~mm}$ You measure the resistivity of the wire in the following way: You connect one end of the wire to one terminal of a battery that has emf $12.0 \mathrm{~V}$ and negligible internal resistance. To the other terminal of the battery you connect a point along the wire so that the length of wire between the battery terminals is $d$. You measure the current in the wire as a function of $d$. The currents are small, so the temperature change of the wire is very small. You plot your results as $I$ versus $1 / d$ and find that the data lie close to a straight line that has slope $600 \mathrm{~A} \cdot \mathrm{m} .$ What is the resistivity of the material of which the wire is made?

Key Concepts

Data Analysis and Linearization

Plotting experimental measurements such as current versus the inverse of the length (1/d) can linearize the relationship between current and resistance. This technique allows one to extract material properties, such as resistivity, by relating the slope of the linear fit to the underlying physical constants and geometric factors.

Resistivity

Resistivity is an intrinsic property of a material that quantifies its opposition to the flow of electric current. It is independent of the sample's shape or dimensions and appears in the formula R = ?(L/A), linking it directly to electrical resistance and the geometry of the conductor.

Ohm's Law

Ohm's Law is a fundamental principle in electrical circuits that relates voltage, current, and resistance through the formula V = IR. This relationship provides the basis for analyzing how changes in resistance or applied voltage will affect the current in an electrical circuit.

Electrical Resistance

Electrical resistance is the measure of how much a material opposes the passage of electric current. For a uniform conductor, resistance is given by R = ?(L/A), where L is the length of the conductor, A is its cross-sectional area, and ? is the resistivity of the material.

Geometric Factors in Resistance

The resistance of a conductor is influenced by its physical dimensions, specifically its length and cross-sectional area. A conductor with a longer length or a smaller cross-sectional area will have a higher resistance. For wires with circular cross sections, the area is calculated using A = ?r², which directly ties the geometry of the wire to its electrical resistance.

Transcript

00:01 Hi, in the given problem, radius of the wire loop is given as r is equal to 2 .00 cm or we can say this is 2 .00 into 10 dash per minus 2 meter.

00:26 Resistance of the wire used in making this loop is r is equal to 0 .600 o.

00:36 On, initial magnetic field is 3 .00 tesla after which the magnetic field varies with the time as a function of time as b is equal to v0, e -res the power minus t by tau, where the value of this tau is 0 .5502.

01:02 In the first part of the problem we have to find the expression, first of all, for the magnetic flux linked with the coil, then with the help of that magnetic flux, the emf induced in the coil, and after which we have to find current passing through this coil.

01:25 So first of all, the magnetic flux linked with the coil will be given as phi is equal to b .a, which becomes b, a, cos theta.

01:45 Now, as initially the plane of oil is perpendicular to the magnetic field.

01:53 So angle between area vector and magnetic field is 0 degree.

01:57 So we can say this is b, a, cos zero degree, and cos zero degree is 1.

02:03 So it comes out to be b into a only.

02:07 And we know the value of the function of b is given as b0 into e minus t by 0 .5 for the value of tau.

02:20 Then for area this is pi into r square.

02:25 So finally this flux here comes out to be for pi r squared.

02:35 This is 2 into 10 dash bar minus to the whole square for b not this is 3 tesla then e ratio power minus t by 0 .5 if we calculate it putting the value of this pi also here we get 5 is equal to 3 .768 into 10 dash to bar 6 8 into 10 dash to bar minus 3, e -r -r -r -pire minus t by 0 .5 weber.

03:28 As this is the magnetic flux, so we'll be having a unit of weber.

03:38 Now, with the help of faradies laws of electromagnetic induction, the expression for emf is given as negative of time derivative of the function of magnetic flux, means minus d5 by dt.

04:10 So it becomes, if we differentiate it here, this is a constant, we remain as it is, minus 3 .768 into 10 dash per minus 3.

04:23 Then differentiation of e to the power x is same as e to the power x means e to the power minus t by 0 .5.

04:31 Then differentiation of this power of e, which comes out to be just minus 1 .0 .5.

04:43 Finally it becomes emf induced is equal to 7 .536 into 10 dash per minus 3...

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