The figure shows a system of two concentric spheres of radii $r_1$, and $r_2$ and kept at temperatures $T_1$ and $T_2$, respectively. The radial rate of flow of heat in a substance between the two concentric spheres, is proportional to :
To measure the radial rate of heat flow, we have to go for integration technique as here the area of the surface through which heat will flow is not constant.
Let us consider an element (spherical shell) of thickness $dx$ and radius $x$ as shown in figure. Let us first find the equivalent thermal resistance between inner and outer sphere.
Resistance of shell $=dR=\frac{dx}{K\times4\pi x^{2}}$
$\begin{bmatrix}from\,R=\frac{1}{KA}\,where\\ K \rightarrow thermal\, conductivity\end{bmatrix}$
$\Rightarrow \int\,dR=R=\int ^{r_{2}}_{r_1} \frac{dx}{4\pi\,Kx^{2}}=\frac{1}{4\pi\,K}\left[\frac{1}{r_{1}}-\frac{1}{r_{2}}\right]$
$=\frac{r_{2}-r_{1}}{4\pi\,K\,\left(r_{1}r_{2}\right)}$
Rate of heat flow $= H$
$=\frac{T_{1}-T_{2}}{R}= \frac{T_{1}-T_{2}}{r_{2}-r_{1}}\times 4\pi K\left(r_{1}r_{2}\right) \propto \frac{r_{1}r_{2}}{r_{2}-r_{1}}$
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Top Questions on electrostatic potential and capacitance
The potential of a point is defined as the work done per unit charge that results in bringing a charge from infinity to a certain point.
Some major things that we should know about electric potential:
They are denoted by V and are a scalar quantity.
It is measured in volts.
Capacitance
The ability of a capacitor of holding the energy in form of an electric charge is defined as capacitance. Similarly, we can also say that capacitance is the storing ability of capacitors, and the unit in which they are measured is “farads”.
The capacitor is in Series and in Parallel as defined below;
In Series
Both the Capacitors C1 and C2 can easily get connected in series. When the capacitors are connected in series then the total capacitance that is Ctotal is less than any one of the capacitor’s capacitance.
In Parallel
Both Capacitor C1 and C2 are connected in parallel. When the capacitors are connected parallelly then the total capacitance that is Ctotal is any one of the capacitor’s capacitance.
