State Biot-Savart's law .Apply Biot savart's law to derive magnetic field at the centre and the
axis of a circular loop.
Please answer
Answers
Answer:
Answer
The Biot-Savart Law is an equation that describes the magnetic field created by a current-carrying wire, and allows you to calculate its strength at various points.
Here,
I= current in the loop
R= radius of the loop
X= distance between O and P
dl= conducting element of the loop
According to the Biot-Savart law,
Magnetic Field at a point P is,
dB space equals space fraction numerator straight mu subscript straight o over denominator 4 straight pi end fraction space fraction numerator straight I space vertical line dl space straight x space straight r vertical line over denominator straight r cubed end fraction
straight r squared space equals space space straight x squared space plus space straight R squared
Since space dl space and space straight r space is space perpendicular comma space
vertical line dl space straight x space straight r vertical line space equals space straight r space dl
Therefore comma
dB space equals space fraction numerator straight mu subscript straight o over denominator 4 straight pi end fraction. space fraction numerator straight I. space dl over denominator left parenthesis straight x squared plus straight R squared right parenthesis end fraction
Again,
dB has two components: dB_x and dB_y
dB_y is cancelled out and the x-component remains.
Therefore
dB subscript straight x space equals space dB space cos space straight theta
cos space straight theta space equals space fraction numerator straight R over denominator left parenthesis straight x squared plus straight R squared right parenthesis to the power of bevelled 1 half end exponent end fraction
therefore space dB subscript straight x space equals space fraction numerator straight mu subscript straight o straight I space dl over denominator 4 straight pi end fraction. space fraction numerator straight R over denominator left parenthesis straight x squared space plus space straight R squared right parenthesis to the power of bevelled 3 over 2 end exponent end fraction
Thus, summation of dl over the loop is given by:
straight B subscript straight x straight i with hat on top space equals space fraction numerator straight mu subscript straight o space straight I space straight R squared over denominator 2 space left parenthesis straight x squared plus straight R squared right parenthesis to the power of bevelled 3 over 2 end exponent end fraction space straight i with hat on top
For Nturns, B=
2(x
2
+R
2
)
3/2
μ
o
NIR
2
Magnetic field lines due to a circular current carrying coil is,
Explanation:
The Biot-Savart Law is an equation that describes the magnetic field created by a current-carrying wire, and allows you to calculate its strength at various points.
Here,
I= current in the loop
R= radius of the loop
X= distance between O and P
dl= conducting element of the loop
According to the Biot-Savart law,
Magnetic Field at a point P is,
dB space equals space fraction numerator straight mu subscript straight o over denominator 4 straight pi end fraction space fraction numerator straight I space vertical line dl space straight x space straight r vertical line over denominator straight r cubed end fraction
straight r squared space equals space space straight x squared space plus space straight R squared
Since space dl space and space straight r space is space perpendicular comma space
vertical line dl space straight x space straight r vertical line space equals space straight r space dl
Therefore comma
dB space equals space fraction numerator straight mu subscript straight o over denominator 4 straight pi end fraction. space fraction numerator straight I. space dl over denominator left parenthesis straight x squared plus straight R squared right parenthesis end fraction
Again,
dB has two components: dB_x and dB_y
dB_y is cancelled out and the x-component remains.
Therefore
dB subscript straight x space equals space dB space cos space straight theta
cos space straight theta space equals space fraction numerator straight R over denominator left parenthesis straight x squared plus straight R squared right parenthesis to the power of bevelled 1 half end exponent end fraction
therefore space dB subscript straight x space equals space fraction numerator straight mu subscript straight o straight I space dl over denominator 4 straight pi end fraction. space fraction numerator straight R over denominator left parenthesis straight x squared space plus space straight R squared right parenthesis to the power of bevelled 3 over 2 end exponent end fraction
Thus, summation of dl over the loop is given by:
straight B subscript straight x straight i with hat on top space equals space fraction numerator straight mu subscript straight o space straight I space straight R squared over denominator 2 space left parenthesis straight x squared plus straight R squared right parenthesis to the power of bevelled 3 over 2 end exponent end fraction space straight i with hat on top
For Nturns, B=
2(x
2
+R
2
)
3/2
μ
o
NIR
2
Magnetic field lines due to a circular current carrying coil is,