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biot savart law:applications & examples

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What is the definition of biot savart law?

“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 } }  }

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