In proton-beam therapy, a high-energy beam of protons is fired at a tumor. The protons come to rest in the tumor, depositing their kinetic energy and breaking apart the tumor’s DNA, thus killing its cells. For one patient, it is desired that 0.10 J of proton energy be deposited in a tumor. To create the proton beam, the protons are accelerated from rest through a 19MV potential difference.

What is the total charge of the protons that must be fired at the tumor to deposit the required energy?

q =E/U =0.1/19•10⁶ =5.26•10⁻⁹ C

Well, it seems like we've got some explosive protons on our hands here! Let's calculate the total charge needed to get the job done.

First, we need to find the work done by the electric field in accelerating the protons. The work done is equal to the potential difference multiplied by the charge:

Work done = potential difference * charge

We can rearrange this equation to solve for charge:

Charge = Work done / potential difference

In this case, the work done is equal to the energy deposited in the tumor, which is 0.10 J. The potential difference is given as 19 MV, which is 19 million volts, or 19 million joules per coulomb.

Plugging in the values:

Charge = 0.10 J / 19 MV

Now, we need to be a little careful with the units. We need to convert 19 MV to joules. Since 1 volt is equal to 1 joule per coulomb, we can simply multiply 19 MV by 1 million, since there are 1 million volts in a megavolt.

Charge = 0.10 J / (19 million joules/coulomb)

Now, let's do the math:

Charge ≈ 0.10 / 19 million coulombs

Hmmm, I don't think I'm capable of calculating that number off the top of my clownish head. But it seems like our protons are going to require a fairly large charge to get the job done.

To find the total charge of the protons that must be fired at the tumor, we can use the formula:

Q = E / V

Where:
- Q is the total charge of the protons
- E is the energy deposited in the tumor (0.10 J)
- V is the potential difference (19 MV = 19 * 10^6 V)

Plugging in the values, we get:

Q = 0.10 J / (19 * 10^6 V)

Q = 5.26 * 10^(-9) C

Therefore, the total charge of the protons that must be fired at the tumor to deposit the required energy is approximately 5.26 * 10^(-9) Coulombs.

To find the total charge of the protons required to deposit the desired energy, we need to use the equation:

\( q = \frac{E}{V} \)

Where:
- q is the total charge of the protons,
- E is the desired energy to be deposited (0.10 J in this case),
- V is the potential difference through which the protons are accelerated (19 MV).

First, let's convert the potential difference to joules by multiplying it by the elementary charge (1.6 x 10^-19 C):

\( V = 19 \, \text{MV} \times (1.6 \times 10^{-19} \, \text{C}) \)

\( V = 30.4 \times 10^{-12} \, \text{C} \)

Now we can substitute the values into the equation to find the total charge:

\( q = \frac{0.10 \, \text{J}}{30.4 \times 10^{-12} \, \text{C}} \)

\( q \approx 3.29 \times 10^{-9} \, \text{C} \)

Therefore, the total charge of the protons that must be fired at the tumor to deposit the required energy is approximately 3.29 nanocoulombs.