a circuit consist of an inductance of 0.05H, a resistance of 20ohms , a capacitor of 100uF and an angle of 100cos200t. find (i) and (q) given that q=0,and i=0 at t=0.

I doubt very much if 100cos200t is an angle. It looks like an applied alternating voltage, with 200t as the "phase angle" .

Recheck the wording of the question.

To find the values of i and q at a given time t, we can use the formulas for current and charge in an RLC circuit.

The current in an RLC circuit is given by the formula:

i(t) = I₀e^(-Rt/2L) * cos(ωt + φ)

where:
- I₀ is the initial current at t = 0
- R is the resistance
- L is the inductance
- ω is the angular frequency (2πf)
- t is the time
- φ is the phase angle

The charge in an RLC circuit is given by the formula:

q(t) = Q₀e^(-Rt/2L) * cos(ωt + φ - π/2)

where:
- Q₀ is the initial charge at t = 0

Given the values:
- Inductance (L) = 0.05H
- Resistance (R) = 20Ω
- Capacitance (C) = 100μF = 100 * 10^(-6) F
- Initial charge (Q₀) = 0 (q = 0 at t = 0)
- Initial current (I₀) = 0 (i = 0 at t = 0)

We need to find the angular frequency (ω) and the phase angle (φ) to substitute into the above formulas.

The angular frequency ω can be calculated as:
ω = 2πf
= 2π (200 Hz)
= 400π rad/s

The phase angle φ can be calculated as:
φ = arctan(2πfRC)
= arctan(2π (200 Hz) (20 Ω) (100 * 10^(-6) F))
= arctan(0.008)

Next, we can substitute the values into the formulas for i(t) and q(t).

i(t) = I₀e^(-Rt/2L) * cos(ωt + φ)
= 0 * e^(-(20Ωt)/(2*0.05H)) * cos(400πt + arctan(0.008))

As i(0) = 0 and q(0) = 0, we can calculate the values of i and q at any given time t using the above formulas.

To find the current (i) and charge (q) in the given circuit, we can use the principles of circuit analysis. First, let's examine the behavior of each component.

1. Inductance (L): The behavior of an inductor is characterized by opposing changes in current. The voltage across an inductor (V_L) can be expressed as V_L = L di/dt, where di/dt represents the rate of change of current with respect to time.

2. Resistance (R): The voltage across a resistor (V_R) is given by Ohm's law: V_R = i * R, where i is the current passing through the resistor.

3. Capacitance (C): The voltage across a capacitor (V_C) is given by the equation V_C = (1 / C) ∫ i dt, where ∫ i dt represents the integral of current with respect to time.

Now, let's proceed with finding the current (i) and charge (q).

Given that q = 0 and i = 0 at t = 0, we can use these initial conditions to solve for the specific values at any given time.

1. To find i, we'll start by examining the inductance component. Given the voltage across the inductor (V_L) as 100cos(200t), we can write the equation V_L = L di/dt.

100cos(200t) = 0.05 di/dt

Separate variables and integrate both sides:

∫ 1 di = ∫ 100cos(200t) / 0.05 dt

Simplifying further:

i = (100 / 0.05) ∫ cos(200t) dt

i = 2000 ∫ cos(200t) dt

Integrate cos(200t) with respect to t:

i = 2000 * (sin(200t) / 200)

So, the current (i) at any given time (t) is:

i = 10 * sin(200t)

2. To find q, we need to examine the behavior of the capacitor. Using the equation V_C = (1 / C) ∫ i dt, we can calculate q.

Given that q = 0 at t = 0, we can integrate the current expression we found earlier:

q = (1 / C) ∫ i dt

q = (1 / 100e-6) ∫ (10 * sin(200t)) dt

Integrating sin(200t) with respect to t:

q = (1 / 100e-6) * (-cos(200t) / 200)

Simplifying further:

q = (-1 / 20) * (cos(200t) / 200)

So, the charge (q) at any given time (t) is:

q = -5e-6 * cos(200t)

Therefore, the current (i) and charge (q) in the circuit are given by the equations:

i = 10 * sin(200t)
q = -5e-6 * cos(200t)

Please note that the units have been omitted in the equations but should be consistent with the given values (e.g., amps for current and farads for capacitance).