Q: A loop that consists of 200 ( N ) turns and an area ( A ) of 0.25 m2 is located in a downward-directed magnetic field ( B ) of 0.40 T. Additionally, the loop’s coils have a resistance ( R ) of 5.0 Ω. If the coils are crushed to an area of 0 m2 in 100 milliseconds ( ms ), what average current ( I ) will be induced in it during the collapse? In what direction will the current flow during this collapse?
A: The following diagram will be used to model the circumstance at hand:
Even though no bar magnet exists in the given scenario, the effect of a downward-directed B-Field is akin to having a stationary permanent magnet’s south pole aimed at the area of interest. Since the magnet is not moving, no opposing B-Field initially exists around the coils in question. Recall, however, that several changing parameters can elicit the rise of an emf. If the magnetic flux ( ɸ ) within the system is stationary, relative motion of the coil’s area in relation to the magnetic flux will create some type of inductive response. This motion could be rotational, and as is the case in our example, it can be created by crushing the loop inward. Let’s review the equations that sums things up regarding induced emfs:
ɸ = BA cos θ
And,
Ɛ = – NΔɸ/Δt
The angle between the magnetic lines of flux and the area of the loop do not change, so the cosine θ function in the flux equation above is of no interest to us; however, the area of the loop does change due to the coil being crushed. For this reason, we determine flux that is induced within the coil in the following manner:
ɸ = BA
ɸ = ( 0.40 T )( 0.25 m2 )
ɸ = 0.100 Tm2
This quantity of flux elicited in the coil is directly proportional to the decrease in the stationary field of flux as a result of the loop’s area decreasing to zero; therefore, the change of flux in the area of the coil will have a negative value:
Δɸ = – 0.100 Tm2
We must now derive an expression that relates the change of flux to the time that transpired during the change. Fortunately, all we need is to divide the change in flux by the time ( t ) interval that has been provided. When this expression is derived, we multiply it by the number of loops in the coil to determine the value of the induced emf:
Ɛ = – NΔɸ/Δt
Ɛ = – ( 200 )( – 0.100 Tm2 / 0.100 s )
Ɛ = 2.0 x 102 V
The current within the coil is determined via usage of Ohm’s Law:
( V / R ) = I
( 2.0 x 102 V / 5.0 Ω ) = 40 A
Determining the direction in which current flows is a bit tricky. Even though the quantity of flux inside of the loop is of interest to us, the orientation of the field outside the loop is useful in determining the orientation of the lines of flux around it. The components of the field outside of the loop point downward. For this reason, the orientation we’re looking for must have components outside of the loop that are directed upward. When this step is taken using the Right-Hand Rule, we see that the current will move in a clockwise direction around the loop as it is crushed.
RENAISSANCE PHYSICS 