In the circuit shown in the diagram below, two thick wires connect a 1.2 volt ba

Question

In the circuit shown in the diagram below, two thick wires connect a 1.2 volt battery to a Nichrome (NiCr) wire. Each thick connecting wire is 17 cm long, and has a radius of 11 mm. The thick wires are made of copper, which has 8.36e+28 mobile electrons per cubic meter, and an electron mobility of 0.00434 (m/s)/(V/m). The Nichrome wire is 5 cm long, and has a radius of 3 mm. Nichrome has 9e+28 mobile electrons per cubic meter, and an electron mobility of 7e-05 (m/s)/(V/m).

What is the magnitude of the electric field in the thick copper wire?
___V/m

What is the magnitude of the electric field in the thin Nichrome wire?
___V/m

Explanation / Answer

1. Let us first list out what has been given

For NiCr,

lNi = 5e-2m; rNi = 3e-3m; nNi = 9e28 e-s/m3; Mobility,µNi = 7e-5 ms-1/Vm-1

For Cu,

lCu = 17e-2m; rCu = 11e-3m; nCu = 8.36e28 e-s/m3; Mobility,µCu = 0.00434 ms-1/Vm-1

2. We now write the formula for current in terms of drift velocity,also known as the trandport equation

I = nAevd, vd is the drift velocity defined as vd = µE

Remembering Kirchoff's law,

ICu = INi

(nAevd)Cu = (nAevd)Ni

(nAeµE)Cu = (nAeµE)Ni

Rearranging the above eqaution to write interms of our unknowns ECu and ENi

(ECu/ENi) = (nAµ)Ni/(nAµ)Cu

= [9e28 * (3.14*0.0032) * 7e-5] / [8.36e28 * (3.14*0.0112) * 0.00434]

= 1.29e-3

--> (ECu/ENi) = 1.29e-3 (1)

This is the ratio of the electric fields

3. We also know from Kirchoff's voltage formula that the voltages around a node sum to zero,

2VCu + VNi = 1.2V

Here we use the formula for V in terms of Electric field intensity, V = El

2*(El)Cu + (El)Ni = 1.2

We don't know the value of each electric field, but we know the value of the ratio ECu/ENi. Thus we divide each term by ENi

2* ECu/ENi * lCu + lNi = 1.2/ENi

Substituting the known values,

(2* 1.29e-3 * 0.17) + 0.05 = 1.2/ENi

0.0504 = 1.2/ENi

--> ENi = 1.2/0/0504 = 23.8 Vm-1

From (1), ECu/ENi = 1.29e-3

ECu = 1.29e-3 * ENi = 1.29e-3 * 23.8 = 0.0307 Vm-1

--> ECu = 0.0307 Vm-1

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