Question

why is anode negatively charged in an electrochemical cell

Answer 1

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In electrochemical cells, at anode electrons are released by oxidation and the same was accepted at the cathode for reduction of ions. So naturally electron rich electrode is considered (here anode) is considered as negative and electron deficient cathode is given positive sign.

In other words,
The terminal towards which the negatively charged ions move and get oxidised (losing electrons) is termed as the anode. And, the other terminal where the positively charged ions gain electrons and get reduced is called cathode.