It could, if you're texting while driving.
My least favorite TV physician, Dr. Oz, says on his website that "experts have grown concerned about the health implications of heavy exposure—specifically, the radiation that the devices emit." Dr. Oz often offers misleading advice, but this is just plain wrong. Experts know that there is no danger from cell phones.
The World Health Organization (WHO) reports that "[a} large number of studies have been performed over the last two decades to assess whether mobile phones pose a potential health risk. To date, no adverse health effects have been established as being caused by mobile phone use."
The National Cancer Institute at the National Institutes of Health says "there is no evidence from studies of cells, animals, or humans that radiofrequency energy can cause cancer."
What is it about cell phones? Cell phones use microwaves - a form of electromagnetic radiation. The history of this issue goes back at least to an article in the New Yorker in 1989. Paul Brodeur alarmed the country when some epidemiological studies supposedly showed an increase in cancer in homes near power lines. [More about epidemiology later.] The power lines as all electrical currents generate magnetic fields - extremely low frequency radiowaves. Scientists never found any causal link. In fact, when more careful epidemiological studies were conducted, the correlation between exposure to electromagnetic fields and cancer disappeared.
Now for the science lesson. Cancer is an example of a biochemical reaction gone horribly wrong. While not well understood in many types of cancer, whatever happens causes the uncontrolled growth of abnormal cells. Documented cancer-causing agents are perchloroethylene (used in dry cleaning), tobacco smoke, ultraviolet light, viruses, and environmental toxins like aflatoxin in peanut butter.
How can light (electromagnetic radiation) cause these biochemical changes? Each particle of light called a photon has an energy that depends on the color of the light. As one can see from the diagram, light comes in more colors than just red. orange, yellow, green, blue, indigo, and violet. The photon can be absorbed by an atom or molecule. The energy then is used to excite an electron. If the energy is large enough, the electron can be stripped from the atom.
Now I like to tell my physics students that chemistry is the science of electrons. When chemical bonds are rearranged or broken in a chemical reaction, it's the electrons that are being exchanged between the atoms. So when the photon is absorbed, this can cause a chemical reaction - if the energy is large enough.
What colors of light have enough energy and can cause biochemical effects? Roughly speaking, you need ultraviolet light. This is why I wear sunscreen when I golf. What about microwaves? A microwave photon has about 1/100,000 the energy of a UV photon. No chemical reactions here.
You may be wondering about your microwave oven. That certainly cause some chemical reactions, right? Yes, but not through the mechanism described above. In this case, the microwave photon is absorbed by a water molecule, and this makes the water molecule rotate. This added motion translate as added energy to the water making the water hotter.
So can a cell phone cook your brain? No. Through evolution, mammals have a wonderful mechanism for ridding the body of excess heat - the circulatory system. However, we can calculate how much cooking is going on. A typical cell phone emits about 1 Watt of power. In a five minute phone call, this could cause an increase in temperature of 0.1ºC in the brain tissues near your ear.
A Note Regarding Epidemiology: This is an important field in science-based medicine. Epidemiology aims to find relationships between exposure to agents and mortality (death) and morbidity (disease). The link between cigarette smoking and lung cancer was first found by epidemiologists, long before any understanding of the physical causes.
A good study is difficult. The most serious threat to a good study is bias. Bias comes in three forms: 1. Selection bias in which subjects are taking part because of an unknown factor that happens to be associated with the exposure and the effect. 2. Information bias where the information gathered is flawed. A typical source of information bias example is when subjects are asked to remember information. 3. Confounders are variables that correlate with both the exposure and the effect. For example , a confounder in the power line study may have been the neighborhoods where the subjects lived.