Yes, you can apply for the GATE EE Exam after completing your B.Tech in Electrical and Electronics Engineering.

The Graduate Aptitude Test in Engineering (GATE) is a national level entrance exam conducted by the Indian Institute of Technology (IIT) for admission to various postgraduate programs in engineering and technology. It is also used for recruitment purposes by several PSUs (Public Sector Undertakings) for their engineering positions.

Eligibility Criteria:
To be eligible for the GATE EE Exam, you need to fulfill the following criteria:
1. Educational Qualification: You must have completed a Bachelor's degree in Engineering/Technology/Architecture or equivalent from a recognized university.
2. Age Limit: There is no specific age limit to apply for the exam.
3. Nationality: You can be of Indian nationality or a foreign national.

Exam Pattern:
The GATE EE Exam consists of a single paper with a duration of 3 hours. The question paper consists of two types of questions:
1. Multiple Choice Questions (MCQs): These questions have four choices, and you have to select the correct answer.
2. Numerical Answer Type (NAT) Questions: These questions require you to enter a numerical value as the answer using a virtual keyboard.

The exam is divided into three sections:
1. General Aptitude: This section tests your language and analytical skills.
2. Engineering Mathematics: This section tests your knowledge of mathematical concepts relevant to electrical engineering.
3. Technical Section: This section tests your understanding of core electrical engineering subjects such as electrical circuits, electromagnetic fields, power systems, control systems, etc.

Preparing for the Exam:
To prepare for the GATE EE Exam, you should follow these steps:
1. Understand the Syllabus: Familiarize yourself with the GATE EE syllabus and identify the topics you need to focus on.
2. Study Material: Collect relevant study materials such as textbooks, reference books, and previous years' question papers.
3. Make a Study Plan: Create a study plan that includes regular study hours, revision, and practice sessions.
4. Practice Mock Tests: Solve mock tests and previous years' question papers to get acquainted with the exam pattern and improve your speed and accuracy.
5. Time Management: Practice time management during the exam to ensure you can complete the paper within the given time frame.

Conclusion:
After completing your B.Tech in Electrical and Electronics Engineering, you are eligible to apply for the GATE EE Exam. By understanding the exam pattern, preparing a study plan, and practicing mock tests, you can increase your chances of scoring well in the exam. GATE scores are also accepted by various universities and research institutes for admissions to postgraduate programs, which opens up further opportunities for higher studies and research in the field of electrical engineering.