(a) Obtain an expression for the mutual inductance between a long straight wire and a square loop of side a as shown in Fig.
(b) Now assume that the straight wire carries a current of 50 A and the loop is moved to the right with a constant velocity, v = 10 m/s; calculate the induced emf in the loop at the instant when x = 0.2m. Take a = 0.1 m and assume that the loop has a large resistance.

Solution

Consider, a strip of width dx (of the square loop) at a distance x from the wire carrying current.

Magnetic field due to current carrying wire at a distance x from the wire is

B = \frac{μ_{0} I}{2 π x}

Small amount of magnetic flux associated with the strip

dϕ = B dA = \frac{μ_{0} I}{2 πx} (a dx)

Magnetic flux linked with the square loop

ϕ = \frac{μ_{0} Ia}{2 π} \int_{x = b}^{x = a + b} \frac{dx}{x} = \frac{μ_{0} Ia}{2 π} {[\log_{e} x]}_{x = b}^{x = a + b}

ϕ = \frac{μ_{0} Ia}{2 π} \log_{e} (\frac{a + b}{b}) = \frac{μ_{0} Ia}{2 π} \log_{e} (\frac{a}{b} + 1)

Since, Flux,

ϕ = MI

we have

MI = \frac{μ_{0} Ia}{2 π} \log_{e} (\frac{a}{b} + 1)

M = \frac{μ_{0} a}{2 π} \log_{e} (\frac{a}{b} + 1)

which is the required expression for the mutual inductance between a long straight wire and square loop of side a .

b) Given,
Current carried by the straight wire, I = 50 A
Velocity, v = 10 m/s
x = 0.2 m
Thus,
Induced emf in the loop

e = B l v = (\frac{μ_{0} I}{2 πx}) l v = \frac{4 π \times 10^{- 7} \times 50}{2 π \times 0.2} \times 0.1 \times 10 = 5 \times 10^{- 5} volt

Some More Questions From Electric Charges and Fields Chapter

a) An electrostatic field line is a continuous curve. That is, a field line cannot have sudden breaks. Why not?
b) Explain why two field lines never cross each other at any point?

Figure shows tracks of three charged particles in a uniform electrostatic field. Give the signs of the three charges. Which particle has the highest charge to mass ratio?

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Electric Charges And Fields

Some More Questions From Electric Charges and Fields Chapter

a) An electrostatic field line is a continuous curve. That is, a field line cannot have sudden breaks. Why not?
b) Explain why two field lines never cross each other at any point?

Figure shows tracks of three charged particles in a uniform electrostatic field. Give the signs of the three charges. Which particle has the highest charge to mass ratio?

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

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For Daily Free Study Material Join wiredfaculty

Electric Charges And Fields

Some More Questions From Electric Charges and Fields Chapter

a) An electrostatic field line is a continuous curve. That is, a field line cannot have sudden breaks. Why not?b) Explain why two field lines never cross each other at any point?

Figure shows tracks of three charged particles in a uniform electrostatic field. Give the signs of the three charges. Which particle has the highest charge to mass ratio?

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

a) An electrostatic field line is a continuous curve. That is, a field line cannot have sudden breaks. Why not?
b) Explain why two field lines never cross each other at any point?