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Human Body
Electric Shock
Problem Set Solution
Below is a schematic picture of a ground fault circuit interrupter (GFCI), a device installed on
outlet in bathrooms, kitchens, outside and near swimming pools to help protect you against harm from
electric shock. A magnetic core surrounds the ‘hot’ (ungrounded) and ‘neutral’ (grounded) wires in your
outlet and a sense coil is wrapped around the magnetic core. Under safe conditions the current flowing
through the hot wire is the same as the current flowing through the neutral and the magnetic field is
zero in the sense coil. An imbalance of currents could be because a person touched the hot wire while
also touching grounded plumbing so that some of the current flowing from the hot wire flows through
the person instead of through the neutral wire. When an imbalance of currents results there will be a
non zero magnetic field inside the sense coil. A change in magnetic field in the sense coil from zero to a
non zero value in a certain amount of time results in a change in magnetic flux through the sense coil.
This change in magnetic flux through the sense coil induces a voltage in the sense coil and is used by
other electronics to open a switch (interrupter) so that current can no longer flow.
University of British Columbia Department of Physics and Astronomy
Problem 1:
Suppose you are trying to design a GFCI and you want the GFCI to detect a 264 mA imbalance in current
occurring within 25 ms. Your sense coil will have 100 turns, wrapped around a magnetic core made of
iron (99.8% pure) with inner diameter of 1 cm and a thickness of 3 mm diameter. Iron (99.8% pure has a
relative magnetic permeability of  /  0  5000 [1]. What induced emf in the sense coil would the
electronics that interrupts the current have to be activated by?
Solution 1:
When current I passes through a straight wire the magnitude of the magnetic field a distance d away
I
where the B- field points tangent to a circle according to the right2 d
hand rule. The magnetic permeability  is determined by the material that occupies the space where
from the wire is given as B 
you want to determine B . The magnetic field inside the sense coil is the sum of the magnetic fields
from the hot and the neutral wire inside the sense coil. The magnitude of this field is



I
I
 ( I hot  I neutral )
B  Bhot  Bneutral  hot  neutral 
.
2 d
2 d
2 d

Under normal operating conditions when running an appliance, I hot  I neutral and B  0 .

When I hot  I neutral , B  0 .
 
The magnetic flux through one turn of the sense coil is given by  per turn  B  A  BA . We have set

 

B  A  BA because the magnetic field B going through a turn is parallel to the area vector A of the
winding. The flux changes here then changes when the magnitude of the B field changes. When there is
a change of flux in time an emf is induced in the sense coil given by   N
d per turn
dt
where N is the
number of turns in the sense coil.
Calculations:
Substituting the definition of flux in to the formula for  we have   N
dB
A , where on average
dt
dB B B f  Bi


. In this problem
dt
t
t f  ti
 ( I hot  I neutral ) 5000  4 10 7  264 10 3
 5.28 10 2 gauss, Bi  0 and t f  ti  25ms .
Bf 

2
2d
2 (0.5 10 )
i
For N = 100, A   (1.5  10 3 ) 2 we have
University of British Columbia Department of Physics and Astronomy
  100
5.28  10 2
 (1.5  10 3 ) 2 V  1.5  10 3 V.
3
25  10
The electronics needs to be able to react to a 1.5 mV induced voltage in the sense coil.
Without the iron core the induced voltage in the sense coil would be 5000 times less and the electronics
might not be able to react to such a small voltage.
References:
[1] http://info.ee.surrey.ac.uk/Workshop/advice/coils/mu/#mur
Janelle Van Dongen 2010/07/27
University of British Columbia Department of Physics and Astronomy