In a solenoid, a large field is produced parallel to the axis of the solenoid
(in the z-direction in figure 2).
Components of the magnetic field in other directions are cancelled by
opposing fields from neighbouring coils. Outside the solenoid
the field is also very weak due to this cancellation effect and for a solenoid
which is long in comparison to its diameter, the field is very close to
zero. Inside the solenoid the fields from individual coils add together to
form a very strong field along the center of the solenoid.
Figure 1: Magnetic field due to a straight wire
To calculate the magnitude of the field in the solenoid, we used Ampere's law.
Ampere's law relates the circulation of B around a closed loop to the current
flux through the loop x µo.
Figure 2: Magnetic field in a solenoid
This gives the field in the centre of the solenoid.
Note that since the magnitude of the current changes in time, so also
does Bo, i.e., for a sinusoidally varying current
Figure 3: Using Ampere's law to calculate Bo
See Figure 4. |Bo| = µo io N/L is the amplitude (maximum value) of the field. You can also refer to an "average" value of |Bo| called, the root-mean-square (RMS) value. BRMS = |Bo|/sqrt(2).
However, this doesn't tell you what the field outside the solenoid is.
To calculate this you need to use the Biot-Savart law.
From symmetry, along the z-axis all the components of the field
due to a current loop cancel, except the component in the z-direction.
So B at a position z along the axis of the solenoid is given by
Figure 4: Time-variation of the magnetic field in the solenoid, also showing BRMS.
Biot-Savart Lawwhere R = radius of the loop. The 3rd equation shows B as a function of z when z >> R. Note that B decreases rapidly as z increases.
Figure 5: variation of B along the z-axis
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