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In Fig. 29-83, two infinitely long wires carry equal currents i. Each follows a 90°arc on the circumference of the same circle of radius R. Show that the magnetic field Bat the center of the circle is the same as the field Ba distance R below an infinite straight wire carrying a current Ito the left.

Short Answer

Expert verified

Proved.

Step by step solution

01

Given

Figure 29-83.

02

Understanding the concept

In this case, we take the vector sum of the magnetic fields and then use the relation for magnetic field in terms of current, distance, and angle subtended to get the required result. We will use the usual convention as which will represent the direction of the field going into the page and will represent the direction of the field coming out of page

Formula:

Magnetic field due to the circular arc at the center of arc with current (i).

B=μ0i2πRϕ

direction of B is determined by right hand thumb rule

03

 Step 3: Show that the magnetic field B→  at the center of the circle is the same as the field B→ a distance R below an infinite straight wire carrying a current I to the left

Let’s find the magnetic field due to the first and second wire separately. Adding them, we will get the net field due to both wires at the center of the arc using equation 29-9

B1=μ0i4πR+μ0i2πR·π2+μ0i4πR

This field is pointing out of the page.

For wire 2

B2=0+μ0i2πR·π2+0

This field is pointing into the page.

So the net field at the center is

Bc=B1+B2=μ0i4πR+μ0i2πR·π2+μ0i4πR-μ0i2πR·π2Bc=μ0i2πR

We can see that this expression is exactly the same as the equation for the field by a single infinite straight wire (equation 29-4).

Hence, it is proved.

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Most popular questions from this chapter

In Figure, two long straight wires are perpendicular to the page and separated by distance d1=0.75cm. Wire 1 carries 6.5Ainto the page. What are (a) magnitude and (b) direction (into or out of the page) of the current in wire 2 if the net magnetic field due to the two currents is zero at point P located at distance d2=1.50cmfrom wire 2?

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A long wire carrying 100A is perpendicular to the magnetic field lines of a uniform magnetic field of magnitude 5.0 mT. At what distance from the wire is the net magnetic field equal to zero?

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