Trying to homebrew a feat to give my Sorcerer a bit of Wizard flavor

[asorel]

The backstory of my 5th edition Sorcerer character has lead me to try and add some mechanical traits to the character to make him feel more wizard-like. I have attempted this with a homebrewed feat, of which I have two variants at the moment:

Studious Spell Caster Prerequisites: INT 13, Sage background, levels in an arcane spellcaster class

When you take this feat, you receive a spellbook with two Wizard spells of your choice inscribed within it, whose levels may not exceed that of your highest level spell slots. Whenever you find spells in the world, you may copy them into your spellbook in the way a Wizard would, but you do not gain spells every level in the manner of a Wizard. The book may hold no more spells than one quarter of your arcane caster levels, and each day you may prepare a number of spells equal to your INT modifier.

Alternate: No limit to size of book, but aggregate level of prepared spells cannot exceed half of caster level+INT modifier. Meaning, a 4th level Sorcerer with an INT mod of 3 could prepare 5 1st-level spells, a 2nd-level and a 3rd-level, or any other combination whose total levels are not greater than 5.

I've tried to balance these as much as I can, but I would still like to know if this is too overpowered for a feat. If it is, what recommendations are there for balancing it further?

[Ramshack]

Does this imply the Sorcerer can cast the Wizard Spells from his existing sorcerer spells per day, or is this granting the sorcerer additional wizard spells per day he can cast?

[asorel]

Does this imply the Sorcerer can cast the Wizard Spells from his existing sorcerer spells per day, or is this granting the sorcerer additional wizard spells per day he can cast?

I had planned to use the Sorcerer's spell slots, without adding any more.

[Ramshack]

Okay, I was going to say new spell slots would be too powerful. This is still a very powerful feat and I can see the appeal of wanting more spells. Have you looked into the book of darkness pact feature from warlocks that lets them gather ritual spells. That might be something you can base the feat off of as well. It gives you access to a good amount of ritual spells