In the diagram is shown an RL circuit with a switch. ? = 90.0 V , R 1 = 50.0 ? ,
ID: 1448311 • Letter: I
Question
In the diagram is shown an RL circuit with a switch. ? = 90.0 V, R1 = 50.0 ?, R2 = 100.0 ? and L = 60.0 H. Find the values of i1, the current through resistor R1 and i2, the current through resistor R2, the current through the switch, the potential difference across R2, the potential difference across L and the rate of change of the current di2/dt in the time just after the closing of the switch.
a) What is i1 just after the switch is closed?
b) What is i2 just after the switch is closed?
c) What is the value of the current in the switch just after the switch is closed?
d) What is the the potential difference across R2 just after the switch is closed?
e) What is the the potential difference across L just after the switch is closed?
f) What is the rate of change of the current di2/dt in the time just after the closing of the switch?
g) What is i1 a long time after the switch is closed?
h) What is i2 a long time after the switch is closed?
i) What is the value of the current in the switch a long time after the switch is closed?
j) What is the the potential difference across R2 a long time after the switch is closed?
k) What is the the potential difference across L a long time after the switch is closed?
L) What is the rate of change of the current di2/dt a long time after the closing of the switch?
2Explanation / Answer
Let's start this problem RL circuits, for this we use the law Kirchhoff
i = i1 +i2
left mesh
E - R1 i1 = 0 (1)
external mesh ( sin R1)
E – L d i2/dt – R2 i2 = 0 (2)
Part a) current i1
Calculate with 1
i1 = E/R1
i1= 90/50
i1= 1.8 A
Part b) current i2
Let's use the equation 2
60 d i2/dt + 100i2 = 90
An inductor used in the expression for the current is
IR + L dI/dt = E
has as solution
I =E/R (1- e-t/) = L/R
replace
= 60/100= 0.60
i2 = 90/100 (1-e-t/)= 0.90 (1- e-t/0.6)
just after closing the circuit t = 0
i2 = 0.90 (1 -e0)
i2=0
Part c)
The current in the switch is equal to i1
Part d)
If the current is zero voltage it is also zero
V= 0 in R2
Part e) VL=?
This is the expression for the current change all the time.
In the instant the circuit is closed there is no current in the resistor R2, so all voltage must be the inductor as a counter electromotive force
VL= -90V
Part f) di2/dt=?
i2 = 0.90 (1- e-t/0.6)
di2/dt = 0.9 (-1/0.6) (-e-t/0.6)
di2/dt = 1.5 e-t/0.6
This is the expression for the current change all the time.
Again if you just close the circuit t=0 — > e0=1
di2/dt = 1.5
Part g)
Current i1 is through resistor R1 and is not time dependent
i1= 1.8 A
Part h)
i2 expression is found
i2 = 0.90 (1- e-t/0.6)
for a long time after closing the circuit t= inf -- e-inf =0
i2 = 0.90 A
Part i)
the current in the switch is i
i = i1+i2
i= 1.8 + 0.9
i = 2.7 A
Part j)
V2 in R2
V2 = i2 R2
V2 = 0.9 100
V2= 90 V
part k)
all voltage V2 is on, so the inductor no voltage
VL= 0
Part L)
The expression is
di2/dt = 1.5 e-t/0.6
si t = inf e-inf= 0
di2/dt = 0