Charges on capacitor after full charge by battery

Question

When the charging is finished what will be the charge on an ideal parallel plate capacitor ? To be specific what are the charges on different faces of the 2 plates ?

All the books say, after 100% charging, the final charge on capacitor is $$ Q = CV $$

But as far as I think the net charge on the capacitor is 0. So what do we actually mean when we say that the capacitor is having a charge $Q$. I mean, what would the charges on the plates be and most importantly why ?

A Brief explanation on how these charges are coming up on these plates will be highly appreciated. Also, I have searched for it in not only books but also Wikipedia and some SE related sites like Hyperphysics etc. But whatever information I have gained is confusing me as I find some answers contradicting. This is my last attempt to understand the concept.

I have checked these, but were not really helpful : Charging a capacitor (terminals) , Charging a capacitor

Thank you.

Answer 1

By capacitor charge is meant the absolute value of the charge on each capacitor plate: $ \mid Q \mid$. If the battery generates the potential difference $V$ and you connect the capacitor to the battery through a conducting wire, as shown in your picture, once the equilibrium is reached each plate of the capacitor will have a charge $ Q = CV $, where $C$ is the capacitor capacitance.

In your case: $ q_2 = \mid Q \mid $, since that plate is connected to the battery positive pole and $ q_3 = - \mid Q \mid $, since that plate is connected to the negative pole. Finally $ q_1 = q_4 = 0 $ because the electric field generated by the battery will drive the electrons to the positive pole and as far as possible from the negative one.

So as you said the net charge on the system will be zero. The charging process can happen because the battery keeps a constant potential and thus an electric field in the conducting materials (wire and capacitor plate).

Answer 2

$$V = {v_r}(t) + {v_c}(t) = i(t)R + {1 \over C}\int\limits_{{t_0}}^t {i(\tau )d} \tau $$ Differentiating wrt t: $$RC{{di(t)} \over {dt}} + i(t) = 0$$ Solving differential equation: $$I(t) = {V \over R}{e^{{t \over {{\tau _0}}}}}$$ From this equation we notice initially at t = 0 $$I = {V \over R}$$

As time is increasing current starts decreasing until at $t = \infty $ the current in the circuit becomes 0 and at this time $t = \infty $ the capacitor gets 100% charged. But in reality t never reaches $\infty $ and thus, at the limit point(from calculus) the charge never goes |Q| exactly but it certainly remains closest to |Q|, $\forall $ (t$ \le \infty $). By |Q| I mean charge on the single plate of the capacitor.

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Drawing parallelism with the diagram in question, suppose the dielectric slab completely fills the spaces between plates a charge ${q_1}$=+Q is given to the positive plate and ${q_4}$=-Q to the negative plate of the capacitor. The electric field polarizes the dielectric so that the induced charges ${q_2} = - {Q_p}$ and ${q_3} = + {Q_p}$ appear on the faces of the slab.

Electric field at a point between the plates due to charges $ + Q$and$ - Q$ on capacitor plates is ${E_0} = {\sigma \over {{\varepsilon _0}}} = {Q \over {A{\varepsilon _0}}}$ in a direction left to right.

Magnitude of induced charge:$${Q_p} = Q(1 - {1 \over K})$$ where K is dielectric constant. Or $$\overrightarrow E = {{\overrightarrow {{E_0}} } \over K}$$ where $\overrightarrow E $ is resultant electric field.For vacuum, there is no polarisation and hence $\overrightarrow E = \overrightarrow {{E_0}} $ and K=1.