What is the current in a wire of radius R=3.40 mm if the magnitude of the current density is given by (a)Ja=J0r/Rand(b)Jb=J0(1-r/R), in which ris the radial distance and J0=5.50×104A/m2? (c) Which function maximizes the current density near the wire’s surface?

Short Answer

Expert verified

a) Current in a wire when the current density is given by Ja=J0r/Ris1.33A

b) Current in a wire when the current density is given byJb=J01-r/Ris0.666A

c) The function that maximizes the current density near the wire’s surface is Ja.

Step by step solution

01

The given data

a) Current density,J0=5.50×104A/m2

b) Radius of the wire,R=3.40mmor3.40×10-3m

02

Understanding the concept of the flow of current and its density

The term "current density" refers to the quantity of electric current moving across a certain cross-section. We use the relation between the current and the current density to find the current in the wire. After finding the current for different current densities, we can check which function maximizes the current density.

Formulae:

The equation of the current flowing through a small area,i=J.dA ...(i)

The cross-sectional area of the circle, A=πr2 ...(ii)

03

(a) Calculation of the current in a wire

We have, the given value of the current density as:Ja=J0r/RforJ0=5.50×104

The differential cross-sectional area value using equation (ii) can be given as follows:

dA=2πrdr

Substituting these above values in the equation (i), we can get the contained current within the width of the concentric ring assuming Jis directed along the wire with varying radial distances from to r=0tor=Ras follows:

i=J0rR2πrdr=2πJ0Rr=0r=Rr2dr=2πJ0Rr330R=2πJ0RR33=2πJ0R23=2π5.5×104A/m33.40×10-3m23=3.99483A=1.33A

Hence, the value of the current of this case is 1.33 A.

04

(b) Calculation of the current in a wire

We have, the given value of the current density as: Jb=J01-rRforJ0=5.50×104for

The differential cross-sectional area value using equation (ii) can be given as follows:

dA=2πrdr

Substituting these above values in the equation (i), we can get the contained current within the width of the concentric ring assuming Jis directed along the wire with varying radial distances from r = to r = R as follows:

i=J01-rR2πrdr=2πJ0rdr-1Rr2dr=2πJ0r22-1Rr330R=2πJ0R22-1RR33=2πJ0R26=2π5.5×104A/m23.40×10-3m36=3.99486=0.666A

Therefore, the value of the current flow is 0.666 A.

05

(c) Calculation of the function that maximizes the current density near the wire surface

Current through the wire when the current density is Jbis different from that in part (a) because Jbis higher near the center of the cylinder, where the area is smaller for the same radial interval and it is lower outward, resulting in a lower average current density over the cross section and consequently, a lower current than that in part (a). Hence,ja has its maximum value near the surface of the wire.

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