It is now believed that protons and neutrons (which constitute nuclei of ordinary matter) are themselves built out of more elementary units called quarks. A proton and a neutron consist of three quarks each. Two types of quarks, the so called 'up' quark (denoted by $u$) of charge $+(2 / 3) e,$ and the 'down' quark (denoted by $d$) of charge $(-1 / 3) e$ together with electrons build up ordinary matter. (Quarks of other types have also been found which give rise to different unusual varieties of matter.) Suggest a possible quark composition of a proton and neutron.

A proton has three quarks. Let there be n up quarks in a proton, each having a charge of $(+2 / 3)e$.

Charge due to $n$ up quarks $=\left(\frac{2}{3} e\right) n$

Number of down quarks in a proton $=3-n$

Each down quark has a charge of $-\frac{1}{3} e$

Charge due to $(3- n )$ down quarks $=\left(-\frac{1}{3} e\right)(3-n)$

Total charge on a proton $=+e$

$\therefore e=\left(\frac{2}{3} e\right) n+\left(-\frac{1}{3} e\right)(3-n)$

$\Rightarrow e=\frac{2 n e}{3}-e+\frac{n e}{3}$$\Rightarrow 2 e=n e \Rightarrow n=2$

Number of up quarks in a proton, $n=2$ Number of down quarks in a proton $=3-n=3-2=1$

Therefore, a proton can be represented as '$uud$'.

A neutron also has three quarks. Let there be $n$ up quarks in a neutron.

Charge on a neutron due to $n$ up quarks $=\left(+\frac{2}{3} e\right) n$

Number of down quarks is $3-n$, and each having a charge of $-\frac{1}{3} e$.

Charge on a neutron due to $(3-n)$ down quarks $=\left(-\frac{1}{3} e\right)(3-n)$

Total charge on a neutron $=0$

$\therefore 0=\left(\frac{2}{3} e\right) n+\left(-\frac{1}{3} e\right)(3-n)$

$\Rightarrow 0=\frac{2 n e}{3}-e+\frac{n e}{3}$$\Rightarrow e=n e \Rightarrow n=1$

Number of up quarks in a neutron, $n=1$ Number of down quarks in a neutron $=3-n=2$ Therefore, a neutron can be represented as '$udd$'.