how to pass a null value to a foreign key field?

Just allow column university_id of table user to allow NULL value so you can save nulls.

CREATE TABLE user
(
   uid INT NOT NULL,
   Name VARCHAR(30) NOT NULL, 
   university_ID INT NULL,    -- <<== this will allow field to accept NULL
   CONSTRAINT user_fk FOREIGN KEY (university_ID)
       REFERENCES university(university_ID)
)

UPDATE 1

based on your comment, you should be inserting NULL and not ''.

insert into user (name,university_id) values ('harjeet', NULL)

UPDATE 2

$university_id = !empty($university_id) ? "'$university_id'" : "NULL";
insert into user (name,university_id) values ('harjeet', $university_id);

As a sidenote, the query is vulnerable with SQL Injection if the value(s) of the variables came from the outside. Please take a look at the article below to learn how to prevent from it. By using PreparedStatements you can get rid of using single quotes around values.

  • How to prevent SQL injection in PHP?

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