1. Consider an electromagnet lift system (figure not drawn to scale) shown in Figure 1:
Iron core
Iron bar
$l_1 = 6$ cm
$l_2 = 1$ cm
$l_3 = 1$ cm
$l_4 = 13$ cm
Figure 1
Note: Question No. 1 continues on page 2.
EE6503
The iron core is excited by a 100-turn coil (i.e., N) over the central arm (not shown in Figure 1). The cross-sectional area of the iron core and iron bar is 1 cm$^2$. The relative permeability and the density of the iron material are 2300 and 7.8 g/cm$^3$, respectively. The permeability of free space is $4\pi \times 10^{-7}$ H/m. The acceleration due to gravity is 9.81 m/s$^2$.
(a) Find the equivalent reluctance (i.e., R) of the system expressed in terms of the airgap length (i.e., x).
(b) By considering the definition of inductance L, show that
$L(x) = \frac{N^2}{R(x)}$ H
Hence, find the inductance L expressed in terms of the airgap length for the conditions described in part (a).
(c) By considering the field energy stored in the system, show that the magnetic force $f_m$ exerted on the iron bar is given by
$f_m = \frac{1}{2} i^2 \frac{dL(x)}{dx}$ N
where i is the current flowing in the coil. Hence, find the magnetic force expressed in terms of the current and airgap length.
(d) Based on the results obtained in part (c), find the current required to provide a magnetic force to lift the iron bar such that the airgap length is fixed at 5 mm (i.e., x = 5 mm). Hence, or otherwise, find the field energy stored in the system.
