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What is the difference between Ampere’s Law and Biot-Savart’s Law?

What is ampere’s law?

According to this law “The line integral of magnetic field B along a closed path due to current is equal to the product of the permeability of free space and the current enclosed by the closed path”.

Mathematically it is expressed as:

amp

Where

μ0=permeability of free space

i=current flowing through the conductor.

Proof:ampere's law

Consider a straight conductor in which current i is flowing.The current produces the magnetic field B around the conductor.The magnetic field lines are in the form of concentric circles.

Ampere showed that the flux density B at any point near the conductor is directly proportional to the current i and inversely proportional to the distance ‘r’ from the conductor ,so:der of amp lawamp

Where  is the length of the path called circumference of the circle.

Divide the circle representing the magnetic field line into large number of small elements each of length dl.The quantity B.dl is calculated for each element as:

B.dl=Bdlcos= Bdlcos0=Bdl

For complete circle:

amp

Integral form of ampere law:

integral form of ampere law

Differential form of ampere law:

Since the integral form of ampere’s law is:

diff form of ampere law

The above relation is known as differential form of ampere’s circuital law.

Biot savart law examples problems and application:

“The magnetic induction at any point produced by current element is directly proportional to the product of the current and the differential element and inversely proportional to the square of distance of the point from the differential element.”

Formula of Biot Savart law:

\dB=\frac { { \mu  }_{ 0 } }{ 4\pi  } \frac { ids\times \hat { r }  }{ { r }^{ 2 } } \\ Integrating\quad the\quad above\quad relation\\ B=\frac { { \mu  }_{ 0 } }{ 4\pi  } \int { \frac { ids\times \hat { r }  }{ { r }^{ 2 } }  }

Derivation of Biot Savart law:

Biot savart law diagram

To explain the Biot Savart law,we consider a point near a wire carrying current i.Let ds be an element of length of wire and dB be the magnetic induction produced by this length element at a distance r from the current carrying wire.It is found that:

1:The magnitude of magnetic induction dB is proportional to the current i.

dB ∝ i

2:The magnitude of magnetic induction dB is directly proportional to the current dsSinθ

dB ∝  dsSinθ

3:The magnitude of magnetic induction dB is inversely proportional to r².

dB ∝ 1/r²

By combining the above three relations we have:

\\\ \\ \\ dB=\frac { { \mu  }_{ 0 } }{ 4\pi  } \frac { idsSin\theta  }{ { r }^{ 2 } } ---------(1)

The above relation is the differential form of Biot Savart law.In vector form it is given as:

\dB=\frac { { \mu  }_{ 0 } }{ 4\pi  } \frac { ids\times \hat { r }  }{ { r }^{ 2 } } \\ Integrating\quad the\quad above\quad relation\\ B=\frac { { \mu  }_{ 0 } }{ 4\pi  } \int { \frac { ids\times \hat { r }  }{ { r }^{ 2 } }  }

What is ampere's law? According to this law “The line integral of magnetic field B along a closed path due to current is equal to the product of the permeability of free space and the current enclosed by the closed path”. Mathematically it is expressed as: Where μ0=permeability of free space i=current flowing through the conductor. Proof: Consider a straight conductor in which current i is flowing.The current produces the magnetic field B around the conductor.The magnetic field lines are in the form of concentric circles. Ampere showed that the flux density B at any point near the conductor is…

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One comment

  1. muy interesante y fascinante publicación de ciencia física.

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