# How to find the charge at time = t (at any instant)

In summary, the conversation discusses the derivation of the charge on a capacitor using the dot product of magnetic field and area. The resulting equations are used to find the voltage and charge on the capacitor at different time intervals. The conversation also mentions an arbitrary assumption for the value of ##\phi## and asks for alternative methods for the question. The 4th option is unclear and requires further clarification.

Homework Statement
A circuit consists of a coil with inductance ##L## and an uncharged capacitor of capacitance ##C##. The coil is in a constant uniform magnetic field such that the flux through the coil is ##\phi##. At time ##t = 0##, the magnetic field was abruptly switched off. Let ##\omega_0= \frac{1}{\sqrt(LC)}## and ignore the resistance of the circuit. Then,

1) Magnitude of charge on the capacitor is ##|Q(t)|= 2C \omega_0\phi\sin(\omega_o t)##

2) Magnitude of charge on the capacitor is ##|Q(t)|= C \omega_0\phi\sin(\omega_o t)##

3) Initial current in the circuit is infinite.

4) The cyclotron frequencies of all the particles are same.

It is a more than one correct type, with the answers being 2 and 4.
Relevant Equations
Charge on Capacitor ## Q = CV ##
Emf induced in inductor ##V (or E) = -\frac{d\phi}{dt}##
I was not able to derive the charge on the capacitor. But then, I arbitrarily assumed ##\phi=B.A## (Dot product of Magnetic field and Area)

Then, proceeding as follows,

##\phi=BA\cos(\omega_0 t)##
##\frac{d\phi}{dt}=−BA\omega_0\sin(\omega_0 t)##

Now at ##t=0, \phi=BA\cos(0)=BA##
Therefore,
##\frac{d\phi}{dt}=−\phi\omega_0\sin(\omega_0 t)##

Now,
##V(t)=−\frac{d\phi}{dt}##
##V(t)=\phi\omega_0\sin(\omega_0 t)##

And finally,
##Q(t)=CV(t)##
##Q(t)=C\phi\omega_0\sin(\omega_0 t)##
Which corresponds to option 2.

Now, since I arbitrarily assumed the value of ##\phi##, I don't know if it is correct.
Also, I don't understand the 4th option. Can you help in that as well?

Thank You.

Last edited:

PeroK said:
Sorry for that, I just corrected them!

## 1. How do you calculate the charge at a specific time?

To find the charge at a specific time, you need to know the current flowing through the circuit at that time. You can then use the formula Q = I * t, where Q is the charge in Coulombs, I is the current in Amperes, and t is the time in seconds.

## 2. Can the charge at a specific time be negative?

Yes, the charge at a specific time can be negative. This typically occurs in circuits with alternating current where the direction of the current changes over time. In these cases, the charge will be negative when the current is flowing in the opposite direction.

## 3. What units are used to measure charge?

The SI unit for charge is the Coulomb (C). However, in some contexts, charge may also be measured in units of electrons (e). One Coulomb is equal to approximately 6.24 x 10^18 electrons.

## 4. How do you graph the charge over time?

To graph the charge over time, you can plot the charge (y-axis) against the time (x-axis) using a line or bar graph. You can also use a scatter plot if you have discrete data points. Make sure to label your axes and include a legend if needed.

## 5. Can the charge at a specific time change?

Yes, the charge at a specific time can change. This is because the charge is influenced by the current, which can change over time. Additionally, if there are any external factors such as resistance or capacitance in the circuit, they can also affect the charge at a specific time.