**Cerenkov Radiation II**

Part A = θ = cos^-1(c/(nv))

Part B = aerogel

Part C = v = 0.995 c

Part D = 0.97 c

**Solution Below:**

When a charged particle passes straight through a medium faster than the local speed of light, it will emit Cerenkov radiation in a cone. Let’s see how the cone angle is correlated to the speed of the particle.

Part A

Express your answer in terms of v, c, and n.

θ = cos^-1(c/(nv))

Part B

- diamond (n = 2.417)
- crown glass (n = 1.52)
- ice (n = 1.3)
- aerogel (n = 1.03)
- vacuum (n = 1)

The aerogel is the only material that will keep light at >= 0.98c

aerogel

Part C

Express your answer as a multiple of c to three significant figures.

This one has the potential to be confusing. Let’s go through step by step. Start by finding the angle you’ll get for a particle traveling at the speed of light (use the formula from Part A):

θ = cos^-1(c/(nv))

θ = cos^-1(2.998 * 10^8 / (1.03 * 2.998 * 10^8))

θ = cos^-1(0.970873786)

θ = 13.86243 degrees

As the speed of the particle decreases, this angle will also decrease. If we want to find the maximum *detectable* velocity of a particle traveling below the speed of light, we subtract the instrument’s resolution from this angle. Since the instrument has a resolution of 1.2 degrees, we want to find the velocity at 12.66243 degrees (note: either use our calculator or be VERY careful that yours has the correct units – e.g. make sure you’re not inadvertently switching between degrees and radians. Use one or the other, don’t mix them):

12.66243 = cos^-1(c/(nv))

12.66243 = cos^-1(2.998 * 10^8 / (1.03 * v))

cos(12.66243) = (2.998 * 10^8 / (1.03 * v))

0.9756784837 = 291067961 / v

v = 298232644.5 m/s

Divide by the speed of light and you get:

v = 0.995 c

v = 0.995 c

Part D

_{min}that a charged particle can have and still emit Cerenkov radiation in the aerogel?

Express your answer as a multiple of c to two significant figures.

Remember that Cerenkov radiation is emitted when a particle moves faster than the *local* speed of light in a medium. For instance, light moves slower in water than it does air. Water’s refractive index is about 1.33, so light moves 33% slower: 0.667 c. This means if a particle in water is moving faster than 0.667 c, it will emit Cerenkov radiation. The problem told us that the aerogel has a refractive index of 1.03. So just divide 1.0 c by 1.03, which gives you 0.97 c. Above this speed, a particle traveling through the aerogel will emit Cerenkov radiation.

0.97 c

Part D says:

What is the lowest velocity vmin that a charged particle can have and still emit Cerenkov radiation in the aerogel?

Express your answer as a multiple of c to two significant figures.

Thanks! We added a solution (0.97 c) to that part.

is there a potability that part d will be added soon?

Our version of the problem didn’t have a Part D. What does it ask?