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Question
State Ampere’s circuital law.
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Solution 1
Statement: The line integral of the magnetic field `(vecB)` around any closed path is equal to μ0 times
the total current (I) passing through that closed path.
`:.ointvecB.vec(dl)=mu_0I`
Solution 2
Statement: The line integral of the magnetic field `(vecB)` around any closed path is equal to μ0 times
the total current (I) passing through that closed path.
`:.ointvecB.vec(dl)=mu_0I`
Solution 3
Ampere’s law states that the path integral or line integral over a closed loop of the magnetic field produced by a current distribution is given by `oint vec("B") . vec("dl") = µ_0"l"`
where I refers to the current enclosed by the loop.
Ampere’s law is a useful relation that is analogous to Gauss’s law of electrostatics. It is a relation between the tangential component of magnetic field at points on a closed curve and the net current through the area bounded by the curve.
To evaluate the expression for `oint vec("B") . vec("dl")` let us consider a long, straight conductor carrying a current I, passing through the centre of a circle of radius r in a plane perpendicular to the conductor.
According to Biot-Savart law of magnetism, the field has a magnitude `(µ_0"l")/(2pir)` at every point on the circle, and it is tangent to the circle at each point.
The line integral of `vec("B")` around the circle is
`oint vec("B") . vec("dl") = oint(µ_0"I")/(2pir)"dl" = (µ_0"l")/(2pir) oint"dl"`
Since `ointvec("dI") = 2pir` is the circumference of the circle,
Therefore , `oint vec("B") . vec("dl") = µ_0"l"`
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