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Figure 29-62 shows, in cross section, two long straight wires held against a plastic cylinder of radius20.0cm. Wire 1 carries currentrole="math" localid="1663151041166" i1=60.0mAout of the page and is fixed in place at the left side of the cylinder. Wire 2 carries currenti2=40.0mAout of the page and can be moved around the cylinder. At what (positive) angleθ2should wire 2 be positioned such that, at the origin, the net magnetic field due to the two currents has magnitude80.0nT?

Short Answer

Expert verified

The angle θ2is104°

Step by step solution

01

Identification of given data

  1. The radius of the plastic cylinder R=20.0cmor0.200m
  2. Wire 1 carries current i1=60.0mAor60.0×10-3A
  3. Wire 2 carries current i2=40.0mAor40.0×10-3A
  4. The net magnetic field due toi1andi2isB=80.0nTor80×10-9T
02

Understanding the concept

When two parallel current-carrying conductors are placed near each other, they exert a force on each other, either attractive or repulsive, depending on the direction of the current. The fields are expressed by Biot-Savart’s law. It is stated as,

dB=μ04πids×r^r2

Here, dBis the field produced by the current element idsat a point located at rfrom the current element.

Using Biot-Savart’s law and Lorentz force equation, we can calculate the force on the current carrying conductor by another current carrying conductor.

By using, Equation29-4we can find the magnitude of themagnetic fields.Also, we have to find the net componentsB2xandB2yofB2. Now, by usingPythagoras theorem and substituting the given quantities, we can find the angleθ2.

Formulas:

From , 29-4the magnitude of the magnetic field is given by

B=μ0i2πR

Here, Bis magnetic field, μ0is permeability constant, iis current, Ris the distance between the point and the current conducting wire.

03

Determining the angle θ2

From,equation29-4the magnitude of the magnetic field is given by

B=μ0i2πR

By the right-hand rule, the field caused by wirecurrent, evaluated at the coordinate origin, is along+yaxis.

The field caused by the current of wire2's will generally have both an xand ay component which are related to its magnitudeB2and sines and cosines of some angle. When wire 2 is at an angleθ, its net components are given by

B2x=B2sinθ2andB2y=-B2cosθ2

By Pythagoras theorem, the magnitude-squared of the net field is equal to the sum of the square of their net x-component and square of their net y-component.

Therefore,

B2=B2sinθ22+B1-B2cosθ22=B12+B22sin2θ+B22cos2θ-2B1B2cosθ2

B2=B12+B22sin2θ+B22cos2θ-2B1B2cosθ2

Since,sin2θ+cos2θ=1

B2=B12+B22-2B1B2cosθ2

Therefore, angleθ2is

θ2=cos-1B12+B22-B22B1B2

Now, we have

B1=μ0i12πR=4π×10-7T·m/A×60.0×10-3A2×π×0.20m=60×10-9T

Similarly,

B2=μ0i22πR=4π×10-7T·m/A×40.0×10-3A2×π×0.20m=40×10-9T

With the requirement that the net field has magnitudeB=80nT, we find

θ2=cos-1B12+B22-B22B1B2=cos-160×10-9T2+40×10-9T2-80×10-9T22×60×10-9T40×10-9T=cos-1-0.25=104°

Therefore, the angle θ2is104°.

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Most popular questions from this chapter

Equation 29-4 gives the magnitude Bof the magnetic field set up by a current in an infinitely long straight wire, at a point Pat perpendicular distance R from the wire. Suppose that point P is actually at perpendicular distance Rfrom the midpoint of a wire with a finite length L.Using Eq. 29-4 to calculate Bthen results in a certain percentage error. What value must the ratio LRexceed if the percentage error is to be less than 1.00%? That is, what LRgives

BfromEq.29-4-BactualBactual100%=1.00%?

Question: In Fig 29-76 a conductor carries6.0Aalong the closed path abcdefgharunning along 8of the 12edges of a cube of edge length 10cm. (a)Taking the path to be a combination of three square current loops (bcfgb, abgha, and cdefc), find the net magnetic moment of the path in unit-vector notation.(b) What is the magnitude of the net magnetic field at the xyzcoordinates of(0,5.0m,0)?

Figure 29-30 shows four circular Amperian loops (a, b, c, d) concentric with a wire whose current is directed out of the page. The current is uniform across the wire’s circular cross section (the shaded region). Rank the loops according to the magnitude of B.dsaround each, greatest first.

Two wires, both of length L, are formed into a circle and a square, and each carries currenti. Show that the square produces a greater magnetic field at its center than the circle produces at its center.

The magnitude of the magnetic field 88.0cmfrom the axis of a long straight wire is7.30μT. What is the current in the wire?

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