The given figure shows a long straight wire of a circular cross-section (radius a)carrying steady current I. The current I is uniformly distributed across this cross-section. Calculate the magnetic field in the region r < a and r > a.
Question
The given figure shows a long straight wire of a circular cross-section (radius a)carrying steady current I. The current I is uniformly distributed across this cross-section. Calculate the magnetic field in the region r < a and r > a.
Solution
The magnetic field due to a long straight wire carrying a current can be calculated using Ampere's Law, which states that the integral of the magnetic field (B) around a closed loop is equal to the permeability of free space (μ0) times the current (I) enclosed by the loop.
- For r < a:
We consider a circular path of radius r (where r < a) around the wire. The current enclosed by this path is a fraction of the total current I. If the current is uniformly distributed, the enclosed current (I') is proportional to the area of the cross-section of the wire enclosed by the path.
The total area of the cross-section of the wire is πa², and the area enclosed by the path is πr². Therefore, the enclosed current I' = I * (πr² / πa²) = I * (r² / a²).
Applying Ampere's Law, we have:
∮B.dl = μ0 * I' B * 2πr = μ0 * I * (r² / a²) B = μ0 * I * r / (2πa²)
- For r > a:
For r > a, the entire current I is enclosed by the circular path. Applying Ampere's Law, we have:
∮B.dl = μ0 * I B * 2πr = μ0 * I B = μ0 * I / (2πr)
So, the magnetic field inside the wire (r < a) is directly proportional to r, while outside the wire (r > a), it is inversely proportional to r.
Similar Questions
A long straight wire of circular cross section of radius a carries asteady current I. The current is uniformly distributed across itscross section. The ratio of the magnitudes of magnetic field at apoint distant a/2 above the surface of wire to that at a point distanta/2 below its surface is(i) 4 :1(ii) 1:1(iii) 4: 3(iv) 3 :4
A long cylindrical wire (radius = 4 cm) carries a current of 22 A that is uniformly distributed over a cross-section of the wire. What is the magnitude of the magnetic field at a point that is 0.1 cm from the axis of the wire?
Which of the following pattern correctly describes the magnetic field around a long straight wire carrying current?[1 Mark](a) straight lines perpendicular to the wire.(b) straight lines parallel to the wire.(c) radial lines originating from the wire.(d) concentric circles centred around the wire.
A long wire with a square cross-section 15 cm on a side carries a current of 15 A that is uniformly distributed over the cross-section of the wire. What is the value of the integral ∮𝐵⋅𝑑𝑠 along a square with side length 6 cm concentric with the wire and with sides parallel to those of the wire?
A conducting w ire carrying a steady current I is shaped as shown in thefigure below. All connected straight segments meet at right angles. Whatis the magnetic moment of the current loop?I abbaˆiˆjˆk(a)Iab (ˆj + ˆk)(b)Iab (ˆj − ˆk)(c)√2Iab (ˆj + ˆk)(d)Iab (ˆk − ˆj)11
Upgrade your grade with Knowee
Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.