The best way to do this is to have the tcpdump command added to to the sudo configuration and set up so as to not require a password.
This can be done by adding a like to end of the sudoers file (or a file in sudoers.d, depending on your distribution) that looks something like this
myUserID ALL = (root) NOPASSWD: /usr/sbin/tcpdump
You will need to adjust the username and path to tcpdump as appropriate. Note also that this needs to go AFTER any other rules that might apply to you, since it will be treated as an exception to the normal policy.
Once done, invoke tcpdump as
sudo /usr/sbin/tcpdump
A note of caution - the sudo configuration files must be edited with visudo to prevent a whole world of misery, and only privileged users can change these files.
Please see:-
man sudo
man 5 sudoers
Depending on your distribution, one or the other of these might help
Ubuntu Forums (Ubuntu and similar)
2.3. Configuring sudo Access (RH and similar)