A 1.0μF capacitor with an initial stored energy of 0.50 J is discharged through a 1.0MΩ resistor. (a) What is the initial charge on the capacitor? (b) What is the current through the resistor when the discharge starts? Find an expression that gives, as a function of time t, (c) the potential difference V C ​ across the capacitor, (d) the potential difference V R ​ across the resistor, and (e) the rate at which thermal energy is produced in the resistor.

A) q_o = 0.001 C

B) I = 0.001•e^(-t)

C) V_c = 1000e^(-t)

D) V_r = 1000e^(-t)

E) P = e^(-2t) watts

Explanation:

A) We are given;

Initial stored energy; U_o = 0.5 J

Capacitance; C = 1.0μF = 1 × 10^(-6) F

To find the charge, we will use the formula for energy in capacitors which is given by;

U = q²/2C

Thus, since we are dealing with initial energy, U is U_o and q is q_o

Making q the subject, we have;

q_o = √2CU_o

q_o = √(2 × 1 × 10^(-6) × 0.5)

q_o = 0.001 C

B) The charge as a function of time is expressed as;

q = q_o•e^(-t/RC)

Now the current is gotten by differentiating the charge function. Thus;

I = (q_o/RC)•e^(-t/RC)

Where;

R is Resistance = 1.0MΩ = 1 × 10^(6) Ω

C is capacitance = 1 × 10^(-6) F

(q_o/RC) is the initial current = 0.001/(1 × 10^(6) × 1 × 10^(-6))

(q_o/RC) = 0.001 A

Thus;

I = 0.001•e^(-t/(1 × 10^(6) × 1 × 10^(-6)))

I = 0.001•e^(-t)

C) Formula for potential difference across the capacitor is;

V_c = IR

I = 0.001•e^(-t)

R = 1 × 10^(6) Ω

Thus;

V_c = 1 × 10^(6) × 0.001•e^(-t)

V_c = 1000e^(-t)

D) Potential difference across the resistor will be the same as that across the capacitor because the resistor is connected in parallel to the capacitor.

Thus;

V_r = V_c = 1000e^(-t)

E) rate at which thermal energy is produced is basically the power.

Thus;

P = (V_r)²/R

P = (1000²e^(-2t))/1 × 10^(6)

P = e^(-2t) watts

sleep paralysis is a parasomnia, events associated with sleep.

sleep paralysis happens when you are in the middle of waking up or falling asleep. it is the feeling of being conscious but unable to move. it definitely involves hallucinations.

i had a friend that had sleep paralysis and he told me a story one time where he was laying in bed and he couldn't move his body no matter how hard he tried and there was someone standing in the corner of his room coming towards him. he couldn't move and you can't "wake up" because you are already conscious. sometimes sleep paralysis can be triggered by frequent waking up during the night based off of multiple trips to the bathroom, or if you had ever been abused, sleep paralysis can start happening from things like that.

as technology advances, new experiments often expose problems in accepted theories. c

hope i

-lavira: )