State biot savart law and apply it to find the magnetic field due to a circular loop carrying current at a point a) at its centre b) on the axis
Answers
Biot Savart Law
Biot Savart Law is an equation that describes the magnetic field created by a current carrying wire and allows us to calculate its strength at various points .
According to Biot Savart Law , the magnetic field produced by infinitesimal length of conductor is given by -
where, dL = infinitesimal length of conductor carrying current
1r = unit vector directing from current to field point.
Magnetic field at centre of loop
According to Biot Savart Law -
For the complete loop , integrate along the circumference 2piR.
=> B = [(uo * I)/(4πR²)]∫dL
=> B = [(uo * I)/(4πR²)] * 2πR
=> B = (uo * I)/(2R)
Thus, Magnetic field at centre of loop = (uo * I)/(2R)
where, uo = 4π * 10^-7 T.m/A
Magnetic field along axis of loop
By integrating z component we can calculate magnetic field on axis of loop .
Integrating along circumference of circle
=> Bz = (uo/4π) * [(2πR²I)/(z²+R²)√(z²+R²)]
Thus, Magnetic field along axis of loop is given by -
Bz = (uo/4π) * [(2πR²I)/(z²+R²)√(z²+R²)]
Answer:
Biot Savart Law
Biot Savart Law is an equation that describes the magnetic field created by a current carrying wire and allows us to calculate its strength at various points .
According to Biot Savart Law , the magnetic field produced by infinitesimal length of conductor is given by -
dB = \frac{uo * I * dL X 1r}{4\pi r^{2} }dB=
4πr
2
uo∗I∗dLX1r
where, dL = infinitesimal length of conductor carrying current
1r = unit vector directing from current to field point.
Magnetic field at centre of loop
According to Biot Savart Law -
dB = \frac{uo * I * dL X 1r}{4\pi r^{2} }dB=
4πr
2
uo∗I∗dLX1r
dB = \frac{uo *I * dL * sin\alpha }{4\pi R^{2} }dB=
4πR
2
uo∗I∗dL∗sinα
For the complete loop , integrate along the circumference 2piR.
=> B = [(uo * I)/(4πR²)]∫dL
=> B = [(uo * I)/(4πR²)] * 2πR
=> B = (uo * I)/(2R)
Thus, Magnetic field at centre of loop = (uo * I)/(2R)
where, uo = 4π * 10^-7 T.m/A
Magnetic field along axis of loop
By integrating z component we can calculate magnetic field on axis of loop .
dBz = \frac{uo * I * dL * R}{4\pi{(z^{2} +R^{2}})*\sqrt{z} ^{2} +R^{2} }dBz=
4π(z
2
+R
2
)∗
z
2
+R
2
uo∗I∗dL∗R
Integrating along circumference of circle
=> Bz = (uo/4π) * [(2πR²I)/(z²+R²)√(z²+R²)]
Thus, Magnetic field along axis of loop is given by -
Bz = (uo/4π) * [(2πR²I)/(z²+R²)√(z²+R²)]