Charged Particle in Magnetic Field: r vs E Graph in 59 Sec 🔥

 

❓ Concept

🎬 Charged Particle in Magnetic Field — r vs E Graph in 59 Sec

Charged particle magnetic field mein gaya…
energy badhi…
par sawal yeh hai 👇
👉 radius linearly badhega ya curve banayega? ⚡🤔

Is concept ko pakad liya,
toh graph-based magnetism questions JEE mein free ho jaate hain 🔥

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🖼️ Concept Image

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✍️ Short Explanation

Yeh question calculation ka nahi,
relation samajhne ka test hai.
Bas teen steps yaad rakho —
force → speed → energy.

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🔹 Step 1 — Magnetic Force Causes Circular Motion (FOUNDATION 💯)**

Jab charged particle ki velocity magnetic field ke perpendicular hoti hai:

$$qvB=\frac{mv^2}{r}$$

👉 Magnetic force centripetal force ka kaam karta hai
aur particle circular path follow karta hai.

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🔹 Step 2 — Radius Depends on Speed

Upar wale relation se:

$$r=\frac{mv}{qB}$$

So:

$$\boxed{r\propto v}$$

📌 Intuition:

• Zyada speed ⇒ magnetic field ke liye bend karna mushkil

• Result ⇒ larger radius

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🔹 Step 3 — Link Between Speed and Kinetic Energy

Kinetic energy ka formula:

$$E=\frac{1}{2}mv^2$$

Isse:

$$v=\sqrt{\frac{2E}{m}} \quad\Rightarrow\quad \boxed{v\propto\sqrt{E}}$$

📌 Energy badhne par speed linearly nahi,
square-root ke saath badhti hai.

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🔹 Step 4 — Final Relation: r vs E (MOST IMPORTANT 🔥)**

Ab dono relations combine karo:

$$r\propto v \quad\text{and}\quad v\propto\sqrt{E}$$

So:

$$\boxed{r\propto\sqrt{E}}$$

👉 Radius energy ke square-root ke saath badhta hai
❌ Linear nahi
❌ Decreasing bhi nahi

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🔹 Step 5 — Graph Shape Insight (JEE FAVORITE 🧠)**

Since:

$$r\propto\sqrt{E}$$

Graph ka nature:

• Origin se start karta hai

• Pehle steep rise

• Baad mein slow flattening

• Always increasing

• Concave downward

📌 JEE options mein:

• Straight line ❌

• Upward bending curve ❌

• Decreasing curve ❌

• Square-root type curve ✔️

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⭐ Golden Summary Box

• Magnetic field → circular motion

• Radius depends on speed

• Speed depends on $\sqrt{E}$
  

• $r\propto\sqrt{E}$

• Graph: increasing, concave down

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✅ Final Takeaway

Agar question bole:

“r vs E ka graph choose karo”

👉 Turant yaad karo:

$$\boxed{{\displaystyle r\propto \sqrt{\mathrm{E}}}}$$

Aur square-root shape wali curve tick kar do ✔️
Full marks guaranteed 🚀