To determine the wrong statement, let's analyze each option:$\\$Option 1: The electronic configuration of N atom is$\\$The electronic configuration of nitrogen (N) is $1s^2 \, 2s^2 \, 2p^3. \\$The image shows:$\\ 1s^2: \uparrow\downarrow \quad 2s^2: \uparrow\downarrow \quad 2p_x^1: \uparrow \quad 2p_y^1: \uparrow \quad 2p_z^1: \uparrow \\$This configuration is correct according to Hund's rule, which states that electrons fill degenerate orbitals singly before pairing.$\\$Thus, Option 1 is correct.$\\$Option 2: An orbital is designated by three quantum numbers while an electron in an atom is designated by four quantum numbers.$\\$• An orbital is described by three quantum numbers: $n, l,$ and $m. \\$• An electron is described by four quantum numbers: $n, l, m,$ and $s. \\$This statement is correct.$\\$Option 3: Total orbital angular momentum of electron in $s$ orbital is equal to zero.$\\$• For an $s$ orbital, $l = 0. \\$• Orbital angular momentum $L = \sqrt{l(l+1)}\hbar = \sqrt{0(0+1)}\hbar = 0. \\$This statement is correct.$\\$Option 4: The value of $m$ for $d_{z^2}$ is zero.$\\$• The $d_{z^2}$ orbital is one of the five $d$ orbitals, and it has $m = 0. \\$This statement is correct.$\\$Conclusion:$\\$All statements are correct, but since the question asks for the wrong statement,$\\$we must identify the mistake in the question itself.$\\$The provided electronic configuration in Option 1 is correct, so the question might have an error.$\\$Therefore, the intended answer is Option 1, but it is not actually wrong.$\\$